Official Riftbound ruling

If I play Singularity and my opponent counters it, can I still play Jhin, Meticulous Killer for Rune of this card's domainRune of this card's domain 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.

Riot's question and answer, word for word, from theUnleashed Rules FAQ and Clarifications

The cards

  • Singularity
    Singularity

    Spell

    Deal 6 to each of up to two units.

  • Jhin, Meticulous Killer
    Jhin, Meticulous Killer

    Unit

    Vision (When you play me, look at the top card of your Main Deck. You may recycle it.)

    If you've spent 4 Energy4 Energy or more to play a spell this turn, you may play me for Mind RuneMind Rune.