Singed's passive is based on his mana max. That's why I boost it quite a lot. Actually, his mana is around 4000 (due to masteries), his HP is 2667+ 1000 (due to mana) = 3667 and his AP is 185 + 0.03*4000 = 305 (due to Archangel's Staff) at level 18.

That makes him very strong but very harmful in the same time (poison dealing 70+0.3*305=161 magic damage per sec!).

Have a look, leave comments and enjoy the game.

Early game

In the beginning

The reason why you have to start with boots is because one of Singed's best thing is his movement speed (very interesting while combining with his poison or his "throw"). Moreover, taking boots allow you to take potions (you can also take 2 health potions and 1 mana potion (or else) if you're scared of being out of mana).

Afterwards

You have to get the Rod of Ages. It gives you life, mana (so life as well) and AP. Moreover, buying it in early game gives you the opportunity to get more and more life and mana (so life as well) in late game (because of its passive).

The Force of Nature gives you movement speed, magic resist, and a lot of HP regen (all very interesting for Singed). I usally start with the HP regen part (the 2 Regrowth Pendants). But you can also start with the magic resist (especially if there's a burst AP ennemy a bit fed).

Mid game

Here, you can buy either Archangel's Staff or Lich Bane.
I prefer starting with Archangel's Staff (even though I wrote the opposite in the build). Starting with the Tear of Goddess, it gives you more and more mana (and life! Actually, every spell gives you ~1 HP). Plus it gives you mana regen, useful for the poison (and the other spells of course).
NB : The Tear of Goddess doesn't stack mana while activating Singed's Poison ! (too bad... with 1 sec CD it could have been fun!)

The Lich Bane is not common with Singed but very useful. Indeed, it gives mana (so life!), AP, mafic resist AND movement speed. Those 5 things are all very important for Singed. Let's mention that the Sheen is useful after his slow or his throw. As I said you can also buy this before the Archangel's Staff.

Late game

The Banshee's Veil is always useful (especially if powerful AP ennemies). But what's great with Singed is that it gives life AND mana (so life again!).

If you want some Armor (many AD's in the other team), instead of buying a Banshee's Veil you could get a Thornmail (very powerful object).