Forget playing styles, this guides all the about the builds you can use depending on the team combos. Basically, Singed has 3 core items, Liandry's Torment, Hextech Revolver/an item than builds from it and Rylai's Crystal Scepter. The following builds adds two items that decides what kind of build it. The boots would be sorceror's, but you can also get merc or mobility, the enchantments could be either homeguard or alacrity, but I wouldn't worry about it until end game/ when I have some extra gold left. All runes, masteries and ability sequence are the same (jungle has different masteries).

Ghost is a pretty good spell for Singed for obvious reasons, and ignite makes you deal even more damage over time, so when their teammates say "don't chase the Singed", it's too late anyways. Alternate spells could be flash and teleport, flash for a sudden flash fling, and teleport if you're solo top.

In the jungle build, we build revolver into wraith, and get some armor (sunfire), the last item could be of your choice, but keep is mind mana isn't so useful when you have blue (so frozen heart is a good idea, esp. if they're dps). Even though your w is useless, level it before e because e is for the sake of the cc, might as well make your slow, slow more.

This is personally my favourite build, for it can overlap with other builds. Since Singed already has a use for mana (passive: mana to hp), might as well get TWO uses for it, and because you get stacks by blowing mana, Singed, with his toggle, easily racks stacks. So obvious, your second item should be some mana. I chose frozen heart over rod of ages because the catalyst passive is useless, I have spell vamp and enough mana. I also chose manamune over arch angel because maramune is more effective the more mana you have, so know you can do something other than running around: because you need to wait between attacks anyways, you can attack with awesome maramune damage, run, attack again, this saves times and very effective for experienced players.

The most common build, where all your mana needs comes from of rod of ages. Because rod also gives hp (and hp regen), you can skip the revolver and get something else (preferably defense).

Great power comes great need to now die, so I chose sunfire and abyssal as my two defense items, not only do they provide defense, but offense as well: abyssal to lower their magic resist for your poison, and sunfire because singed really "sticks" to the enemies. The abyssal and sunfire combo are a really common sight, as sunfire deals aoe magic damage, and abyssal lowers their magic resist, making sunfire deal even more damage. It's like having an extra spell!

The classic support item is aegis/runic, so I grab that is I want to go support Singed. Then I grab another item that really benefits myself, Shurelia's. Early game, the philosopher's stone it builds from gives me the useful health regen, and the active is pretty useful for spreading my poison, and flinging. Because now you have no health get, a flask is highly recommended (as your first item). Singed is a viable support, for he has two cc, and can harass well. A good strategy would be to just throw your w, and let your carry harass them under the slow.

All in all, Singed is a highly versatile champion, and can high potential of turning the game around. Personally, I like to jungle with tear of the goddess, for he can easily clear camps and his ganks are godlike. What I do to save time is when a monster's almost finished, I run back and let it chase me into my poison, finishing it off and saving time. When it comes to ganking, I around all the way around and gank from the end of the lane while leaving a trial, and when they run away, they run through my trail anyways, and then I can do my combo. The reason I get tear is because I can turn my q on for 1 sec, get a stack, stop for 4 seconds, repeat, and easily rack up stacks, while being able to afford giving my blue to the mid. My personal full build is the 3 cores, maramune and iceborn gauntlet. The reason for iceborn is because it's mana and cdr in one, plus the passive isn't entirely useless, I can do my w, fling, slow again.