Singed is one of the most hilarious champions in the game. I frequently find myself laughing when I play with him. I find that he scales extremely well in Twisted Treeline. I always build tanky and I still often end up with the most damage on my team. The greatest trick with Singed is that if the enemy pays considerable attention to you, they will almost certainly lose. Of course, if the enemy team is smart they should ignore you entirely. Your job is to be so obnoxious that that is very, very difficult.

As always, build to the enemy team.

Ask yourself who is doing the damage. Even if there are two AP champions and one AD, it could still be the Tryndamere shutting you down, in which case you still need to build armor.
If they are AD heavy, and even if they aren't, Frozen Heart. It will allow you to turret dive to a stupid extent. The CDR is very nice, and the mana will give you health. I almost always get it. It would be worth it even without the passive on Singed, honestly.
Feel free to pick up another Glacial Shroud after FH is complete for 30% CDR and a ****ton more tankiness.
If they are AP heavy, or if your team is, Abyssal Scepter is a great pickup (it will increase damage for all AP champions on your team). Spirit Visage and Quicksilver Sash are also viable.
If they are mostly melee, rush Rylai's for maximum kiting.
If they have lots of CC, definitely get Mercury's Treads. If they have Teemo, or really any stealth anything, Hextech Sweeper is an amazing item on Singed (it's pretty damn good regardless, actually). Rengar will be surprised when he ults and you reveal him immediately with the active.

Pick the second summoner's spell based on your team (always take ghost). If no one has ignite, I will replace exhaust with ignite. In general, exhaust is probably a better pick as it will allow you to get clutch flip off, which will secure a kill from your team far more effectively than ignite, even if you are fighting alone. Cleanse is also viable.

To keep this guide short, I will say most of what I have to say here.

My favorite aspect of Singed occurs the same way pretty much every time I play him: the enemy team gets me to about 10% or less. I run at just the right time with my ult up, they spend much too much time chasing me and trying to kill me, and then Singed friends come and kill them all. Or they kill themselves in my poison. Knowing when to run on Singed is crucial. Ideally you should look extremely killable. Everyone will tell you not to chase Singed, but it is VERY hard to resist chasing a champion with 10, 5, or 1% HP.

High Priority Skills:

Be very defensive until 6. Just farm. You are quite squishy and you have no escape abilities, and not much kill potential either.
Flip is Singed's most difficult ability to master. With it you can help both your enemies and allies escape. You can also aid both your allies and enemies in securing a kill. Consider very carefully will flip will place an enemy.
ALWAYS be the one to engage. I frequently tell my allies that if I engage every team fight, we should win every team fight. The enemy team spending just a few seconds attacking you while your allies jump on them will put them at a ridiculous disadvantage.
Assure that your opponents spend as much time on your Mega Adhesive as possible. Toss it IN FRONT of you when running away, and in front of them when chasing (if possible).
Time your ult such that it lasts the entire duration of a fight. As soon as the **** hits the fan, have it up. You will die ASTONISHINGLY fast without it. Don't get CC'd before it is up.
Run in zig zags in front of your enemies so they can't avoid the poison. Also, poison trail is perfect for killing stealth opponents if you can predict where they will go (or where Teemo is standing).

Other tricks:

Good for lane phase, even before 6: run past your laning opponent with poison trail on, then turn around, flip them, and run back to where you started.
Similarly, if you're actually about to die, all alone, and being chased, simply turn around, flip them, and run in the other direction. It's an excellent juke.
Leave poison on more or less all of the time once you get Catalyst (and especially once you get tear). I have literally never gone oom on Singed after 6.
When your ult is active and you've got even a single armor item, you can basically disregard turrets. Many a time have I gotten away with turret diving the entire enemy team (and running once they're on me, of course. This is an easy way to get them to chase you.)
When your ult is up and you're reasonably built (at least catalyst and tear) get your allies to stand in the bush near the enemy altar while you take it. This will force them to attack you, and that will make them lose.
You run super fast with your ult up. This makes it easy to get really far from your team. Be careful.