Singe in my opinion is one of the most fun to play champions in LoL, you basically just run around free farming and roam in between top and mid lane stealing all the jungle creeps on the way there.

Singe is a champion that is very easy to start playing but somewhat hard to master - His skill set is very simple but it will take a good amount of games to know how much damage you can take and in what match ups you need to think and not just run all over the place laying down poison

You want to start the game with boots / 3 hp pots

If you are laning against teemo or olaf wait in base until 1:35 and buy a 4th health pot before going to lane

On your first shop you will want to pick up Heart of Gold or the start of it ruby crystal and a ward / 1-2 health pots
DO NOT BE GREEDY BASE IF YOU ARE LOW HP

Second shop buy philosopher stone - if you are laning vs Teemo buy ninja tabi before philos

At this point you will start to shop quite frequently because you will now be able to push past the enemies first turret safely and take jungle creeps pick up Catalyst ASAP if you are buying it in pieces buy the mana crystal first

Pick up blasting wand if the enemies tower is still standing purchase 1 ward if the tower is down purchase 2 and take over the enemy jungle (Ward enemy red if you are on blue side enemy blue if on purple side)

After buying rod of ages you now are extremely tanky for mid game and can safely attempt to fight the enemy top lane / jungler 2on1 if you have your ultimate off cd

Keep farming buy Needlessly large rod and proceed to build death cap at this point your main goal changes from farming to picking up kills with your ultimate on you will be able to take roughly 30-40% of an enemy AD carries life in one fling and your poison will start to really chip away at enemies life pools

Save for rylais and once you get it you will be unkillable (IF YOUR ULTIMATE IS UP) unless you are CCed in place and focused by an entire enemy team. If the enemy AP carry is super fed at this point buy Negatron cloak before completing rylais

To complete the build buy Force of Nature then proceed to sell your gold/10 items (always sell philos before heart of gold)
At this point if you are being killed by the enemy AD carry buy Thorn mail. If the enemy team is running two AP heroes buy Abyssal scepter.
If you are having no problems surviving or you are super fed skip FON/Thorn mail and buy Will of the Ancients

The most important part for your rune sets are having flat armor seals and magic penetration marks

Movement speed quintessences are amazing on singe and the best choice but if you are low on IP you don't have to use them Flat AP / magic pen / armor / gold per 5 will all work as well

Glyphs are also very flexible because in the top solo lane you will most likely be vs physical damage heroes and the magic resist is just for mid/late game survival in team fights

If you are vsing the champion jarvan you can take flash over ghost if you are worried about being bullied in lane

ignite is always used in solo queue because it helps ensure that you win lane

if you are playing in a premade team you will probably want to use teleport over ignite to have more control of dragon early game baron late game

For the first few levels you don't want to be pushing the lane with your poison (q ability) you want to attempt to freeze the enemy minions just outside of your tower range so when the enemy champion attempts to last hit the minion you can use fling (e ability) and throw the enemy champ into tower range

After level 5 you will start to want to push the lane with poison

When you hit level 6 it will be safe for you to push past the tower - You use q kill all the minions when your minions reach the enemy tower look at the mini map when you see your own minions about half way in between the outer and inner turret run past the tower and farm the incoming wave then proceed to enter the enemy jungle and farm. If the enemy champ chases you he is going miss out on EXP and CS. If the enenmy champ chases you and catches up to you use ult and run away, try and save ghost as a last resort . NEVER DO THIS WITHOUT A WARD

If you are purple side ward the river near the baron pit

If you are blue side ward the tri bush

When early game comes to a close roughly level 8-10 you should have your second tier of boots (Ninja tabi vs melee merc treads vs AP ) a Heart of Gold and a Philosopher stone and a catalyst. At this point you will want to start roaming the map and picking up farm in other lanes.

For example if mid lane is headed bot to try and score a kill farm the remaining creeps in top lane and run to the middle lane and farm the creeps that are in mid lane before roaming back to top

By 20 minutes into the game you should have at least 170 creeps in a game where you are dominating your lane and have free farm you should have around 210-230

Remember that you are a pusher before a team fighter if there is a team fight about to break out always push your lane before meeting up with your team - AP singe works very well as a clean up hero if you enter the team fight late.

If you master singe you will dominate in lower ELO with this build ( sub 1.9k) If you are higher elo then that the build will change and you will want to build more tanky and build Rylais after rod of ages and possibly skip death cap completely. The reasoning behind this is in lower ratings the enemy AD carry will usually panic if a wild singe appears in his face where as in higher elo the AD carry will stand their ground and proceed to melt you

With this build you are basically going to play solo and give your team the finger you will be able to carry almost any team if you get a few early kills and maintain high CS.

Your main job in team fights will be the initiator using ghost + your ultimate to catch enemies who are out of position and fling them towards your team. Before you have rylais you will not be able to save your AD carry from enemy bruisers and shouldn't worry about peeling after rylais your poison trail will let you team safely kite enemies around

Pros-
Very high burst damage (fling is 1.0 ap ratio)
Probably the best farmer in the game
Very hard to shut down in lane
Massive aoe dmg when you get death cap
AOE slow after rylais
Amazing pusher

Cons
Teemo/olaf will annihilate you in lane
You will go squish in team fights if you don't have your ultimate off cool down