Welcome to my humble singed guide, i main singed, and have realized certain things about the way he is played that simply do not work, my build has been perfected over a period of almost two years, this build is merely a suggestion, you will need your own play style, this is meant more to make you think about how singed is played than anything else.
singed is an ap tank who does tons of damage while sopping up any and all enemy attacks without flinching, he can be classified as a bruiser, or an ap nuke, or all number of other things, he can be a magewick esque character built with ap, and an uber tank built defensively.

i use one rune page universally for singed, he has so much health at the beginning of the game his only problem is ad carries, this is countered by armor, ap and magic pen make you do great early game damage and help with early laning when you are buying health.

There are 2 sets of masteries singed can reasonably use, defense and utility, utility is played in lane, and is AMAZING with the massive amount of mana from the double ROA build i use, thanks to strength of spirit you end up with around 60-70 hp regen per second in the end game. defense is fairly common sense in the jungle.

there are three items i ALWAYS build on singed, two of them are rod of ages, one is merc treads (unless their team has absolutely 0 cc, in which case you can build swiftness, but this is very rare) the reason for the double ROA build is because it is pretty much the most beneficial item on singed, you get hp + mp which transfers to hp so double hp bonus which makes it give you only slightly less HP than a warmogs, additionally it gives you ap to feed the 1:1 ratio on your throw and increase the damage of your poison.

other items are situational
my favorite is force of nature, it makes you faster and gives you more hp regen, only build this if the enemy has either a fed ap user or more than one ap user

second fave is thorn mail, when using the double ROA build you put out a TON of damage making you a good target to focus, thorn mail kills those annoying little ad carries so you don't have to, only build if the enemy has a fed ad carry or lots of constant ad (shaco ez and other ad who focus on abilities do not apply)

deathcap, you have a 1:1 ratio on your throw, added to the fact that with this puppy you can 3 shot squishes with your poison, works best combined with other ap situational items.

rilai's crystal scepter, this one is a bit odd, it is very good on singed but as soon as you get it you lose your ability to bait properly because a slowed enemy tends not to chase as much, if you want to buy this do so when you begin team fighting and stop laning.

atma's is a good item on singed, 4k-5k hp turns into a LOT of ad, use with a nashor's for increased attack speed and more throws (not recommended but can be fun)

basically rank poison as fast as possible, poison has a worse ratio then throw but makes up for it with better scaling, a worthy choice.

jungle singed should start with poison.

if you chose to teamfight at level 1 it is your choice weather to go poison or throw, i suggest throw however, if you head to lane without a teamfight, go with poison, it does a lot more early game damage.

it is your choice weather to rank throw or slow second, i vouch for throw as it gives you massive burst, but the slow can help with ganks and such.

singed is best with ghost, it syncs with his play pirfectly, use ghost + insanity potion to be both uncatchable and inescapable, just try to save it and trigger it along with your ult for maximum effectiveness.

the secondary spell for a lane singed should be teleport, it allows you to push lanes VERY hard, just walk through minion waves on one lane, teleport to the next, walk through, and boom, the pressure is on 2 towers, also effective for getting back to lane in early game and ganking by porting to wards.

the secondary spell for jungle singed should be smite, fairly obvious why.

Creeping with singed is simple, run through the minions with poison on, if you can, make it look like you are going up to throw the enemy champion, this typically makes them run to minimize damage, really it just minimizes the damage you take.

Jungling singed is slightly odd, what you need to do is quickly flick on your poison before running through the camps, then turn it off again before you start auto attacking, repeat when the poison weirs off, this wil conserve mana, make sure the whole camp is poisoned in one run, this can take some practice.

When ganking start by ulting in the position closest to your enemies where they can not see you, immediately run out and throw them, once you have thrown them stay between them and their tower with poison on, do not use your slow until your throw is 2-3 seconds from cooling down, when it hits the 2-3 second mark, place your slow between the enemy and their base, keep them poisoned and auto attack them, when they pass the slow, throw them backwards into the slow again to keep them longer.

Build rod of ages twice, use merc treads unless the enemy has no cc (including 2 or more exausts) and stay between the enemy and the tower with poison on, thats about it, i could do a detailed summary but i did that in the other sections, thank you, and please leave feedback.