Flash is viable vs certain lane/jungle combinations.
Ignite is viable for snowballing certain match-ups.

Items:
Laning:
Dark Seal is an incredibly cost-effective item on Singed, to the point where stacking them, despite the unique passives, is very viable. I typically try to only build one, but if needed, two or more can be built.
Corrupting Potion is also incredibly strong on Singed as an offensive and defensive item, as the DoT starts ticking as soon as an enemy stands in his poison, but doesn't stop ticking until three seconds after his poison expires on them. In other words, if they even touch his poison, Corrupting Potion will do 30-60 bonus magic damage, minimum.
Catalyst of Aeons will provide Singed with an incredible amount of health and sustain in lane. Build it as soon as you can.
Ninja Tabi are almost always the best boots on Singed.
Core Items:
Turbo Chemtank gives Singed incredible lane threat, incredible health and sustain, and excellent scaling into team fights, if he chooses to participate.
Zz'Rot Portal, combined with Turbo Chemtank, makes Singed practically unkillable, and very mobile, while also providing him a means to more directly siege structures. He's great at killing minions, but not too great at siege.
Rylai's Crystal Scepter allows your now incredibly tanky Singed to deal a bit of damage, and stick on people forever, as well as disrupt the enemy team in team fights, run away from just about everything, and deal a bit more damage to boot. Excellent item.
Randuin's Omen is nearly essential if you plan on team fighting.
Thornmail is an excellent choice to back up Singed's massive health pool in team fights and against certain laning opponents while splitting.
Offensive Options:
Liandry's Torment: is the single strongest damage item on singed, in combination with Rylai's Crystal Scepter. Build it if you're snowballing or bringing damage to the team, but not if you're acting as a main tank.
Boots of Swiftness are a solid boot option on Singed. Mobility is his best friend, and if you don't need Ninja Tabi, these are usually your second choice.
Void Staff is another great damage item on Singed, if you wish to go very damage heavy, particularly against a tanky team. You won't typically build this until after Rylai's Crystal Scepter + Liandry's Torment, but the item trio will shred even tanks.
Rod of Ages has fallen out of favor compared to Turbo Chemtank, but it is still a fine rush option, particularly when facing a team that is likely to engage on you, rather than the other way around. It provides a bit more health, and more damage, at the cost of lost engage.
Everfrost is basically Rod of Ages with a faster power spike, and fewer scaling stats. The active allows some additional chase potential and burst, however. Decent for damage heavy builds.
Seraph's Embrace will provide the most late game damage of the mana items on Singed, but has the longest ramp-up. If you're confident that the game is going to go long, consider picking up an early Tear of the Goddess and letting the stacks tick while you scale. The downside is it provides less utility than most of the alternatives such as Everfrost.
Hextech Rocketbelt is basically Everfrost for singed, but cheaper, and with worse stats. The benefit, however, is it does provide slightly higher catch potential with the active, than Everfrost does with the slow.
Iceborn Gauntlet is a strange item on Singed, and isn't a very common build, but it can be used to win lane against certain physical damage champions. It's rarely better than other alternatives.
Defensive Items:
Dead Man's Plate: This will be your go-to over Randuin's Omen any time you're not worried about enemy crit, or are focusing on split pushing.
Guardian Angel: This is an MR heavier alternative to something like Thornmail in the recommended build, if you're planning on doing some heavy team fighting.
Spirit Visage is your item of choice when purely concerned with magic damage. It's a rare build, but gives your healing over time a nice boost.
Banner of Command is a decent option on Singed if you want to pressure opposing lanes at the same time. The stats are much appreciated by him.
Frozen Heart: Singed is a good candidate for this item, as the mana provides him with health, and the CDR, though not essential, is still appreciated. Build it against heavy attack speed comps.
Mercury's Treads: A rare build on Singed. Only pick these up if the enemy team is very CC heavy, and it's preventing you from doing your job very well, be that team fighting or split pushing.
Locket of the Iron Solari: This is your team fighting Magic Resist item of choice. Fairly low build priority.
Banshee's Veil is a great item on Singed statistically, but you'll rarely run into significant enough magic damage to merit it, due to the nature of the champion, role, and strong item choices.