Detail, resolution and timbre are my priorities, along with a correctness of timing, what the French call "vitesse". I will gladly forgo "euphony" in favor of agility, fleetness of foot, so to speak.
The other thing for me is bass energy. High bass levels hurt! I am much more sensitive to this than most. In my experience it is very very difficult to find a speaker that doesn't bloat the bass. (This might not always be the speaker's fault - room acoustics could be the deciding factor. Trouble is, I'm always spending my room treatment budget on a front-end upgrade...)
Of course, this is all subjective! My hearing is not the same as yours, O Gentle Reader. I freely admit that my preferences are idiosyncratic at best.
I have found one "torture test" to be most revealing of a component's quality as pertains to what I like: how it does on densely orchestrated fortissimi. Only the very very best gear will handle those without serious compression and without timbre going straight into the Diaper Genie.
The above is only opinion. But as it is *my* opinion, surely you recognize that it is, definitively, canonically and by default, Correct. (unhinged cackle)
cheers ww