If I play Singularity and my opponent counters it, can I still play Jhin, Meticulous Killer for [C] this turn?

You can. Jhin checks the energy spent to play a spell, not whether that spell resolved. If the energy was spent and the spell became a Finalized Chain Item (and was not undone, for example as part of step 5 of the process of play) then it doesn’t matter if the spell is later countered. The energy wasn’t unspent, even if the spell was countered.