Some good picks would be:
(Q+AA him whenever he tries to CS, especially after you have Sheen. You have sustain, scale well, and can easily set up ganks against him)
(mixed damage, ranged harass, sustain)
(you can easily harass him and if he tries to wither you early, he'll run OOM)
(manaless, shield to absorb damage, good harass)
(it's a farm lane, but you scale just as well as Nasus)
(stand infront of the minions and just hit him whenever he tries to farm, you have mixed damage and will wear him down. It doesn't even matter if he tries to Q farm your ghouls, you should be able to keep him behind in gold and exp)
(you scale just as well, you do a lot of mixed damage, massive sustain in lane)
If you picked Singed first and were countered, I would suggest trading farm until 6 (unless your jungler ganks) and then proxy farming. You might be trading farm with a Nasus, but a properly farmed Singed is usually more useful in teamfights and will hit critical mass sooner. I would recommend getting Rylai's Crystal Scepter, Frozen Heart, and Liandry's Torment in that case. Not the best solution, but it's better than feeding in lane and being underfarmed.