Clairvoyance This spell takes a circular area out of the fog of war where you put it and illuminates it, so you can see the immediate area, people inside it, and people within grass nearby. It does not show up stealthed people. It lasts for 6 seconds unless you take the mastery in it where it lasts for 10. It has a relatively short cooldown, around 40 seconds or so.
This spell has many uses. You almost certainly wont see a ranked game without it it is hugely useful in illuminating areas of the map to set up ganks on isolated champions or observe where people are converging on you from. It can also be used to scout Baron and Dragon, and generally set up a huge amount of kills and prevent many deaths on your team. Typically, the support should/will take it because they have less need of other summoner spells and typically have to have more map awareness, especially on global casters like Soraka. Here is a map showing good places to put clairvoyance to see strategic points, but it is not limited to these locations:

Notably, the key areas to check in passive parts of the game are the major buffs, dragon, baron, and immediately at the spawn to see whos bought what and where they are going. If the enemy has a jungler, it can be used to scout where they are as well throughout the game. It is a spell that can take a while to get used to, but when you have, it is a huge help and your intuition takes over for you on deciding where to scout.
If youre on a team with clairvoyance, and you need it somewhere, just ping and type CV and they ought to know what you mean and will oblige as soon as they can. This spell works especially well in a team of premades on voice communication, and i speak from experience that co-ordination helps significantly.

Clarity Clarity restores an amount of mana to yourself corresponding to your level, and half to those around you unless the mastery in clarity is taken by the user, when it gives the same. Its not the most common spell, unless youre playing on normal where it is abundant. Try not to lane with two claritys if possible, as it wastes a lot of mana and usually renders one of them useless. As for taking it, this is a champion based spell. It is a poor choice on AD carries most the time even if they have mana issues, as it takes up a slot for a defensive or offensive summoner spell. Tanks and supports can benefit from it, but again this is mainly a spell for the supports that duo lane in bot, as they can often spec into the mastery that gives it equal amounts to allies, allowing them to use it with a lot of people in teamfights or just to help their laning partner.

A lot of games simply dont have a clarity AD carries shouldnt be taking it, AP carries could but usually dont, support needs to prioritize CV and if they take both have no defensive mechanism other than their own spells, and tanks cant usually spec into it meaning they cannot help their team very much with it. Also, a lot of the time champions will be building their own mana regen or taking blue, so it really is one of those spells that can be spared from most games. You must base it off your team, I cannot emphasize that enough.

Cleanse A very nice summoner spell, often overlooked on normal games, that removes all CC on you when you activate it and reduces incoming CC by 65% for a short duration after activation. Note however it does not work with all CC, for exaple, it will not help with knockups and pushbacks such as Alistar Headbutt or Lee Sin Dragon's Rage.

It fits on many different champions, simply because it benefits all champions. Tanks can initiate more effectively with it, soaking up more CC, especially used in conjunction with a Banshee's Veil. Carries of all kind can use it to chase or run away for a kill or survival, and supports can take it if they wish for the same reason of survival.
Imagine, for example, you are playing Master Yi, laning with a Maokai against Lux and Taric. You pop your Wuju Style and run towards the weakened Lux. Stunned by Taric, you hammer your Cleanse button, and thus have a halfed stun from Lux afterwards. This all happened losing you about 2 seconds at most, as opposed to 4 or 5. So you then proceed to exhaust Lux and Highlander, and get a kill. You might even refresh abilities and then move onto taric for the doublekill. Ignite, ghost, all those kind of spells doesnt have the same significance.

Thats not to say cleanse is a must, it takes experience to learn when to use it as well as a quick mind and itchy finger. It is also fairly useless against teams with little CC compared to other offensive spells you could take. But cleanse is notably versatile and worth considering if youre unsure of a second skill to get.

Exhaust Slows, reducing damage from auto attacks (70%) and abilities and item damage (35%), such as from Sunfire Aegis. It used to blind and slow, but now Riot changed it so it doesnt completely destroy champions like Tryndamere. However, it basically still does and now its a viable option on absolutely everyone because of the way it reduces ability damage. It can be used offensively to secure kills, or defensively to run away. Lets give another scenario: A teamfight erupts and you are sitting in the middle of a well timed Fiddlesticks Crowstorm from the other team. Smack exhaust on him, and his damage is gimped. The same applies if a Tryndamere had hopped in, critting only for 100 rather than 500, for example. So Riot mostly failed in balancing it, but they did make it viable on more people and in more situations.

Notable people to take it though are AD carries, as it helps so much in a 1v1 situation as well. Try and time exhaust for when they activate their skills using the example of Master Yi again, exhausting him after he uses Highlander and Wuju Style benefits you in a 1v1 scenario, unless of course he runs away through the slow. Often, if you are exhausted, nowadays the best answer is simply to fight through it, as you could still secure a kill whereas running is futile with the heavy slow and just weakens you without weakening your opponent. On an AP carry though, the best answer is to try and hop away with a Flash or ability such as Distortion, and then turn around and nuke after the exhaust is over.

Also worth mentioning is the mastery in exhaust reduces the targets Armor and Magic Resist by 10, which coupled with runes early game can mean close to true damage at the start of the game - Especially with a The Brutalizer or Sorcerer's Shoes.

Flash Another classic good spell in LoL, moving you a small distance instantly towards the location of your cursor. Again, its versatile: Hopping towards an enemy to chase, or away to dodge CC and save your life. The great thing about flash, is although it is such a short range, it can dodge powerful spells that could be fired at you or, better yet, go over walls:

(I made this myself but its terrible so feel free to use it wherever you like. You can flash over all the walls shown but the map is unclear in perspective in areas).
Theres not much more to say. Just make sure you dont waste flash unnecessarily if you can surive or chase otherwise. You can also flash from grass to grass to be very ninja in cases. More on flash will come in other sections, such as gank escape.

Fortify Grants invulnerability to turrets, and 50% more attack speed for 6 seconds, when the mastery is taken these attacks splash a small amount as well. By no means a bad take, despite how many people in their guides simply refuse it. This is because other spells are usually a lot better for carries and supports as I have mentioned before. However, its very beneficial for tanks who can spec into the mastery as they go, and only really need one core spell. Lets go back to scenarios (yay) for how useful it is:

You are aced, after a poorly executed teamfight by you. Its ~35 minutes into the game so death timers are harsh. The two remaining ememies, Gangplank and Kennen, start working on your outer turret which is miraculously still there and destroy it. The tank pops fortify when they reach the inner and save you a turret on full health, because you have all respawned by the time their next wave can arrive. Tank saves the day!

Fortify isnt always needed, but again, if a tank takes cleanse they may see it more viable to get fortify as well because the team doesnt need helping with other spells like heal. Its all dependent. Another good use of fortify is when you are tower dived, or when enemies try to tank towers and the double damage tears them apart, and they usually dont see it coming. This works very well with moves like Power Fist, although prepare to be accused of being a noob, although deep down, they're just very angry :).

The final and rarely noticed use is that the fortify mastery grants 9 bonus damage against minions when it is on cooldown. This helps because tanks tend to be poor last hitters anyway as a broad generalization, allowing ease of farming in laning. Specifically though, it helps with jungling tanks like Amumu or Shen, speeding up the jungle which saves a lot of time, especially for a slow jungler like Shen, allowing early ganks and hopefully kills.

Ghost Another bread and butter spell. Ghost, when activated, speeds you up and negates unit collision, and lasts a bit longer and goes a bit quicker when specced into. Another versatile spell, it helps in chasing and running away, through mobs of minions as well to obstruct an aggressor. It also combos nicely with other champion abilities like Dr. Mundo Maximum Dosage and Master Yi Highlander. The sustained speed buff makes it more useful over spells like flash in the laning phase in many circumstances, as there are few walls to overcome. Due to its versatility, every and any character can take it to suit their own needs, such as initiation for tanks like Rammus that may want an extra speed buff with Powerball to taunt enemies to allow the catch up of his team. Theres not much more to add in the way of strategy, but there a hundreds of examples of how they can be used. The only thing really worth mentioning is that the negated unit collision means you can kite (dragging the enemy when they cannot hit you) the opposition by running through waves of minions to aid your escape or fight.

Heal Another spell relatively uncommon in all games. It can be useful in many scenarios early and midgame, saving peoples lives and sustaining them longer in fights to get more damage down, or defensively when being chased or turret dived. The reason its so uncommon though is because by late game or even as early as mid game, the effectiveness is noticeably decreased and negligible in really long games, and it fills up a slot that could be far more useful in something like exhaust, or cleanse especially considering that late game aces etc really are the aspect that defines the eventual winner of the game.

Another important aspect to consider when using heal is its timing - heal grants a debuff that reduces the amount of healing you recieve a short time after healing. Maybe, therefore, if a Janna Monsoon is healing you and your team, the heal should wait until afterwards or else you'll find you're getting less heal. People like Dr. Mundo also have to be careful with this, because it reduces the power of his ultimate.

As a result, heal is, or should, only really be taken if you think it will give you a sound early game that leads into late game dominance, or else you essentially sacrifice a slot useful in other lategame spells. Dont be discouraged though to take it.

Ignite Ignite is a purely offensive spell dealing damage scaling with level, but it does at least 50 true damage. When levelled, it grants 10 bonus AP when on cooldown. Therefore, you can scale it with runes and a dorans ring, for example, to have substantial early game AP as an AP carry, although usually 15% spell pen and full utility is better through masteries. It serves as a good spell for early game kills, especially when combined with another spell such as exhaust, meaning relatively easy killing if you co-ordinate well with a partner. It becomes less useful late game despite scaling with level, although it still works effectively through midgame and goes very well in conjunction with moves like Children of the Grave.

Similarly with heal, ignite debuffs the target reduces its healing effectiveness. As a result, ignite can be used wisely at the start of teamfights on champions like Dr. Mundo (to use the example again) to substantially weaken his ultimate, rather than saving it to the end of the fight to try and pick someone off. It strongly depends on situation though.

AD champs can take this if they want, but only as a buff to their damage and it often isnt necessary compared to other versatile spells. AP champions also benefit from the damage, and tend to have better escape methods so they can afford to, especially as they sometimes need more kills to keep up on gold and farm. Tanks and support shouldnt really take it compared to the other spells they need defensively or supportingly.

Promote Promote was removed from the game a long while ago now, for reasons such as this;

But it is of course worth mentioning that it no longer exists. Feel free to take it on any character, but if you manage to id be impressed :D.

Rally summons a beacon with 200 + 25 x level health for 15 seconds which increases allied base physical damage by 10-35, and has scaling health with champion level. the mastery in it also grants the those around the totem AP, helping everyone, but especially casters. I have literally never played a game where someone took this skill and used it seriously, so I can only talk off maths here. On tanks only it could be a wise idea, because they have slots to take it as I have explained before, and the AD can benefit the physical carries and would synergize well with Janna and her shield. The advantage is only fairly small compared to teh kind of AD carries will have later game - every little helps is very much the ideology here. It would also require skilled placement in a frantic teamfight to draw attention away from one shotting the totem, and quick thinking, making it tricky in circumstances.

CONFIRMATION NEEDED: A very sneaky way to use it against (gullible) players is to ward bait with it, which is where you plonk the totem down in grass where you think the enemy is to see if they are there, and then initiate on them. Another idea is if YOU are sitting in the grass, place the totem down in plain enemy view and with a tad of luck they will go to destroy it, at which point you and/or your team annihilate them. However, the most common and frustrating use is to place it directly on you when you are being chased, and chances are the person chacing you will misclick and waste time and abilities on the ward. This is easily achieved simply by buying actual wards though, and you could take a defensive spell like flash instead.

Revive Very, very frowned upon. To a large extent this is true, as you shouldnt be dying to take revive in the first place, and it takes up a valuable slot. However, there are benefits. (Notably, the mastery provides 400 extra health for two minutes after using revive - very strong early game)

For example, if your team is aced you can revive to help defend a turret that would be threatened by the team. You also get back into action quicker early game, but timers are short anyway early on and its not a huge difference, especially when you have to wait another 10 minutes before it can be used again. A genuinely good use used to be when people played jungle Evelynn (which she cant now due to nerfing of base stats), where they jungled, died to golem and then resurrected for very fast alternative to porting, and also gaining the mastery bonus of additional health which REALLY helped with ganking. This allowed eve to jungle better as well, which is great as she is a superb ganker. Another good application is on 3v3, where when teamfights occur the map is so small that you can revive and run back to the fight and recontribute, to surprisingly effective results. Probably the best use is on champions like Pantheon and Twisted Fate, who have global ultimates so they can teamfight, die, and then resurrect and jump/port back in, cleaning up any remaining enemies and taking the advantage to push.

Many argue though that it is still a waste over other summoner spells, and most the time, I would agree with them. Certainly, in the conventional situation you probably shouldnt be taking revive.

Smite Almost exclusively a jungling spell. It is nearly always essential because it helps steal and prevent the stealing of buffs, and generally serves as good damage to help early jungling. It maintains quite a lot of use throughout a game, because it can be used to quickly clear minions and super minions and push waves towards turrets even later on. It even serves useful when trying to steal major buffs like dragon or baron with help if you have a ward or clairvoyance to target it and time properly. You can then turn on the enemy with a stolen baron buff and be more liklely to win the teamfight. The only non jungle uses of smite are pushing creep waves a little bit quicker, but the difference really is not worth it over another spell. I would not take this spell unless you are jungling though, as there are simply better picks otherwise.

Take note, however, that you can smite objects such as Heimerdinger turrets, which IS quite a big help in a teamfight if you have smite up. What was funnier is before the Gangplank changes you could also sacrifice your own heimerdingers turrets, which was always fun for the trolls.

Teleport Finally, we come to teleport. Teleport channels for a few seconds, uninterrupted unless by stuns or taunts (as opposed to recall), towards a target building or unit, including wards and mushrooms. It can be used by AD champs like sivir and shaco for backdooring, or to simply get to action, or buy tanks to get quickly to a fight scene. You can also teleport away from a nasty situation to save your life, for example, by juking an enemy out and then porting in a brush behind them before they can DPS you down to zero health. This tactic works so long as they dont have a CC that can interrupt the port. Of course, teleport also can take you to a large farm quickly so you can get back to mid in time for a fight, or vice versa.
It can also be used on a sight ward in grass where people really dont see it coming. By porting to a ward in the laning phase behind an enemy, you can pop out of a brush next to them which they thought was perfectly safe, and proceed to kill them with your teammates. Theres lots of possibility again here.

This is to the MOBAFire users that helped me with the guide, give them respect in turn! :)

Xaiolo for the Veteran recommendation
Canoas for help on cleanse
Trediem for specific help across multiple spells