Regarding build:
If enemy laner is a threat I usually grab Chain Vest (against AD) or Negatron (against AP) right after I get my Dark Seal and Boots of Speed. These can be upgraded later into Deadman/Abyssal. Otherwise, I pass these.
Then, first completed item I'd take Liandry's against a tank (the new passive benefits him quite well, especially in extended fights), or Rylai's against squishies (it's just that good as a rush item because of its cost efficency).
Regarding playstyle: I don't really play Singed Singed-style, and it works pretty well for me. What I mean is that I don't proxy, I don't unnecessarily push, and I use autoattacks in trades early quite a lot. (The key here is not to chase with autoattacks, but to weave autoattacks while backing off, even slightly)
Singed's early trading potential is pretty underrated tbh. In previous seasons it wasn't at all this way, but right now with Aery, Scorch and buffed Ignite he's quite a threat. If the wave is in a good position for you, you can punish your lane opponent by putting poison on top of low health minions. The enemy laner will usually just take it, going for the minion and commiting to a trade, and you should be winning that trade if you weave in autoattacks while the poison is on them. That's why I don't like unnecessarily pushing with him, so I can constantly punish the enemy laner when they're trying to farm.
Obviously many times you auto-push the wave with the poison while trading, so yeah in these cases I'll push the wave asap so it resets before the enemy can freeze it.