Well, glad to see other people enjoying one of my favorite champs
Anyways, on to advice-:
You're going hardcore AP. This is generally lulzy, since your Poison can decimate squishies before they even realize it. HOWEVER, this becomes a serious drawback when you're playing against well co-ordinated teams. Why? They will rain CC from hell on you. Even with your passive, unless you get SOME tankiness, you WILL get killed. Singed is fast and durable, yes, but both don't count for much when you're getting super focused and don't have defenses.
Generally, I find a single RoA enough. However, if you feel you lack the oomph, go and get a Rylai's. That slow on it is unbelievable, and the HP is just gravy. With Fling, Mega adhesive, and a Rylai-Poison, you can pretty much guarantee kills on whoever gets caught.
Boots of Swiftness- Trade them in for Mercury Treads against teams with hard CC (we're talking 2+ stuns, or mega slows, or snares, etc.). Trust me you won't regret it.
Earlier I recommended Ninja Tabi too, but now I find that lackluster. If you're facing a DPS heavy team, Swiftness are better.
Warmogs is just fail. It makes you a meatball, and thats it. While Singed is one of the few Champs who can actually rack it up FAST, I don't think its worth the price. You'd do much better investing in the following items-:
Force of Nature: huge health regen, best magic resist in the game, moar movespeed. Whats not to like?
Banshee's Veil: when the other team starts raining all hell down on you for pissing them off, this is just sweet to piss them off even more.
Guardian Angel: just when they thought they had you...
Frozen Heart: hurray for Armor and mana AND cdr..

Archangel is... okay I guess. I mean it'll take forever to charge up, but even at base charge the AP bonus will be significantly huge (higher than Zhonya's afaik).

Starting items: Ruby crystal? Maybe sapphire would do the trick better, but its upto you I guess. I've tried a lot of different starts, and I generally find Singed to be pretty flexible on this.
Skilling order: Fling first is fine. After that, get Poison trail to level 3 ASAP. Why? Because a level 3 poison trail + fully charged RoA will kill all minions you poison. Just run to a lane, poison, run out. Bam, lane cleared. Next, get Adhesive to level 2. This is because the slow % at level 2 is large enough to usually guarantee a kill if the attack is co-ordinated (level 1 is a little chancy IMO). After that I suggest maxing trail then fling then adhesive.

Masteries- I'd recommend 0/9/21, to get Nimbleness in the defensive tree. This, coupled with a full set of evasion seals should net you 8%+ dodge. Not much, BUT if this procs, you get a 10% speed boost. Thats quite important IMO.

Summoner spells- CLEANSE! How can you ignore it? Its just amazing. People will be like "OMFG Singed STUN HIMMMM!!!" and you're like meh, whatever. Too useful IMO.
Ghost is just fine, pretty much mandatory on Singed.

This is just my opinion. I've been maining Singed for around 70+ games by now. I'm SL 20, but play often with level 30 clanmates. All of what I listed has worked for me. Hope this helps.

I love AA on Singed and I main with him. The 3% conversion is nothing to sneeze at when you have (and you will have) 2K+ mana. That item gives 105 AP, 400 mana and 100 health with something of a continuous increase in mana. It really is not bad.

If you can't get 500-600 AP with Singed then you aren't getting AA. Plain and simple. AP Singed needs AA. Maybe even two.

You won't be hating on AA when I have 718 AP and my Fling does 1000 damage.

At least try this build in a practice game or something. It may convert you to AP Singed.

RoA
Boots of Swiftness
Archangel's Staff
Zhonya's Ring
Archangel's Staff
Lich Bane

Of course you can't farm that much in one game, but I have pulled it off once.

That's 755 AP without runes or masteries.

Even if you can't build Lich Bane you still get 675 AP.

If you were to get, say a Rylai's and an Abyssal Scepter instead of the AAs you would only get 535 AP but you would be spending 485 more gold than with the more powerful build.

I know little more about Singed than watching him used against me - so take this with a grain of salt.

I noticed tons of HP stacking, but no mitigation? Wouldn't something like Abyssal Scepter be fantastic on Singed if you're building AP? Glacial Shroud for the cooldowns, armor, and mana?

I usually get an Abyssal Scepter or other MR item if I get pounded by magic damage. But I don't build him as a tank. To me he's a caster that just happens to have 3.5K health end game.

My core build is
RoA
Boots of Swiftness or Mercs
AA
Zhonya's

If I feel comfortable with massing some crazy AP I go with another AA. If not, I go with Abyssal Scepter. By then it's generally over. Sometimes I go Sunfire Cape if I feel like watching minions explode. Literally.

Usually my Trail does 70 + 200ish and my fling does 300 + 400-500. With a Sunfire Cape it's not really fair but I find strict AP is a better option with Singed.

It's fun to carry the team as Singed.

If you try it, just remember that you are not building him as a tank. And, yes, he is crazy durable still (with a natural end stat of 100 Armor or so and 3K health) but your mitigation is not that of a tank when you play AP Singed.

Just add me in game, i'll give you some tricks and hints there

Singed is one of my 4 most played in ranked, with a win % of 70-80

i usually just get ap situational items then boots of mobility usually drink my potion and ghost i run into the enemies they chase me they dont realize my Q hurts by the time they realize they are slowly dieing i fling mega adhesive then by this time my team usually notices there all low and gets the kills i bascially force them to chase me.