I have come here today to share with you my build for the best tower diver in the league. I'm not going to bother putting anymore in the intro, because I am tired.

Greater Quint of Resilience (2)
Because Singed doesn't have skin, so you'll need to plate him with steel
Greater Quint of Warding (1)
Because maybe the enemy team has more than AD
Greater Seal of Resilience (6)
ROCK SOLID
Greater Seal of Warding (3)
So that Karthus doesn't kill you
Greater Glyph of Potency (9)
Because people are stupid, and they will chase you. These people will account for 90% of your kills.
Greater Mark of Intellect (9)
For tons of damage

Get yourself some ******** movement speed and steel plating.

All of the items that I recommended are specifically to make Singed as tanky as possible, without sacrificing his damage. Seeing as how his passive gives him bonus health based on his mana, Rod of Ages and Archangels are useful for making sure he can tower dive before the game even starts, they also add a lot to his ability power when Archangels gets enough stacks.

Rabadon's Deathcap just ensures that nothing within the same postal code as you will survive

Sunfire Cape needs no explanation

Boots of Swiftness
Boots Boots Boots
Boots

Rylai's Crystal Scepter for INFINITE SLOW

Max out Poison Trail immediately. People are stupid. They will chase you and die because of it.

Fling is good if you find yourself at a party, and a table needs flipping.

Mega Adhesive is useless after the first point spent in it.

Insanity Potion is exactly what it sounds like. A potion makes you insane. However being insane includes the ability to spawn dive and fling the enemy's squishy support into your whole team, it's kind of like a bird feeding its children.

Ignite
Singed needs Ignite because he doesn't have an ability that sets people on fire, plus it goes well with poison trail.
Promote
Because Singed doesn't need a second summoner spell, so he just uses the second slot to get more gold, and steal the carry's kills.

All in all, Singed is a tanky mage carry melee fighter assassin.

I'm not even kidding, he can fill any of those roles.

Pros
Takes at least 4/5 of the enemy team to kill you
High damage output
Afraids of nothing
Insanity Potion is actually just a bottle of your enemy's tears
Classy 10/10 would bang

Cons
Can't turn into a bear
People will try to run away from you, but this is hardly a con because YOU DON'T GIVE THEM A CHOICE

You can feed your own team by flinging an unsuspecting Sona into your team's Darius

The most useful farming method for Singed is to press Q and run into a big crowd of minions. Shortly afterward you will notice that you have gained enough gold to buy your full build.

You can kill people by running from them.

You can go into a team fight with Poison Trail on, press R, and start dancing.
Pentakills ensue