First Purchase
Zzrot Portal- "The Purple Tower Eaters"
I like to rush if I'm ahead or even. This gives a hefty amount of Armor, Mr, and HP regen. The Active allows Singed to be a threat to towers and pressure other lanes.
Rod of Ages- "Finely Aged Item"
Rush if in a slow lane and need to scale. Gives Singed all the stats he wants, but is slow to charge up to be worth it 100% of the time.
Righteous Glory- "Time to use my Vroom Vroom Move"
Rush if you are going in to early teamfights and skirmishes. Gives Singed HP and Mana, HP gained from Mana usage, and a great engage tool.
Core Items
Abyssal Scepter- "Spooky Purple Skull"
Get against 2 or more AP if ahead. Gives AP, MR, and an AoE Magic Damage %Increase, which are all great if you can stay near the enemy to utilize the passive.
Rylai's Crystal Scepter- "Pressure Stick"
A must buy on Singed, gives HP, AP, and a slow on his poison. This makes Singed godly in teamfights, splitting, proxying, and basically everything else.
Liandry's Torment- "Burn baby burn, Singed inferno"
Great item to be built later after Rylai's. Gives a DoT based on current hp%, increasing if the target is "impaired", Rylais Slow,

, and

into

root.
Hextech Protobelt- "Hehe xd"
Decent if bought against a squishy team and gives HP, AP, CDR, and an active that gives you a small dash with burst attached. Not very good, but good enough.
Defensive Items
Banner of Command- "Waving your banner all over the place"
Great since its rework, gives Armor, MR, Mana, CDR, and an active that empowers a minion giving it magic immunity. Very cheap and cost efficient.
Spirit Visage- "
Gives MR, HP, CDR, HP regen, and bonus healing from all sources. Best MR item for Singed, gives more regen to ult and lots of Health.
Thornmail- "Stop hitting yourself"
Highest amount of armor and is the best one for Singed in most scenarios. Gives him the ability to deal most AA champions and great to proxy with, giving manaless clear to minions.
Banshee's Veil- "Pretty Jewelry"
Large amount of MR for a cheap amount, but low HP. Passive is great against a catch comp with lots of high impact CC.
Randuin's Omen- "Shield thing"
High HP and Armor item, and reduces damage of critical strikes. Its active slows around you for 4 seconds. Buy against Yasuo, Vayne, or Ashe.
Deadman's Plate- "Wished GP stayed dead"
HP, Armor, and a passive that gives movement speed stacks by moving and gives bonus physical damage and slows on next attack if the stacks reach 100.