Singed is one of the most overlooked champions in the game. When played correctly, he can not only be a great tank and help your team massively in teamfights with his poison, fling, and AOE slow, but he can also use his ultimate to effectively destroy towers and enemies alike.

Mana runes are great, because of Singed's passive. For each 10 mana you have, you gain 2.5 health as well.

I like to get a mix of Magic Resist, Mana, Health, and AP items. The magic resist, mana, and health speak for themselves. The AP is great for boosting the damage on your Fling and Poison Trail.

I like to get Fling first. It makes it a bit tough to farm creeps until you get level 2, but it's fantastic for getting first blood, especially combined with your Ignite. After that, put one point into Poison Trail and one point into your Slow, then max out Poison Trail and Insanity Potion whenever available. I like to max my Fling after that since it does a decent amount of damage at level 5 (300 base), but depending upon how the battle is going, you may wish to max your slow.

Be wary of ranged DPS characters early game, as you don't yet have enough health to reach them without taking severe amounts of damage. If possible, try to ranged with Teemo or Twitch; Having 2 poisons in one lane can be devastating and keep the enemies towerhugging and constantly needing to use health potions and/or recall.

You can dictate the flow of battle with your poison trail. If nobody is jungling, I would recommend getting Blue Buff around level 4-5. When you have blue buff, you can spam your poison trail and never run out of mana.

Early Game Item Goals:
Full Boots
Rod of Ages
Catalyst the Protector (for Banshee's Veil)

At this point, you should have at least one, maybe two, points in your ultimate and a decent amount of health. This is the point when you can start focusing turrets and teamfights. Your ultimate is epic for taking down turrets, and deals out huge chunks of damage in teamfights since it boosts your AD and AP.

Mid-game Item Goals:
Banshee's Veil
Sunfire Cape

Teamfights and base destruction. Need I say more?

Item Goals:
Finish the build if possible, it usually isn't.