What matters most for me is flexibility, i.e., how easy or feasible is it for me through some simple edits and tweaks make several customizations used in the game.
This is important for me because I tend to re-use the asset in multiple places, e.g., a 3D sword with several "upgrade" levels.
As for the most annoying thing for me is weird format or packing that makes the asset barely usable, like, exporting images in PSD format, or lack of transparency when we the sprites in question needs transparency, transparency when the sprite style is classic 32bit or less, or unrigged organic 3D models that are not exported in its bind pose (I can rig, but please make it easy for us devs to work on top of it).
Lastly, license is very important for me, I only consider using if it allows commercial use and customization (not for resell ofc).