1-Tonal Balance...octave to octave, and note to note within each octave, particularly the midrange. I search for neutrality of tonal balance above all else. Failing that (and all 'phones do fail "absolute neutrality" to some degree), I seek SMOOTHNESS of response. Response variations are FAR easier (for me) to accept if they are smooth (low q) rather than sudden (high q).
2-Frequency extension. There's music down there, folks! True "sub-bass" response to at least 30hz, preferably to 20hz or below IS IMPORTANT! And it's not being a "dumb-bass" either! Most of the "bass" which so impresses "dumb-basses" is actually mid-bass in the 50-100hz octave. That "boom, boom, boom" stuff is not REAL bass! The "real bass" lives in the octave below 50hz! And it's IMMENSELY important not just for the musical content ("notes" and overtones), but for the information conveyed about the acoustic environment in which a recording is made!
3-Low distortion to high volumes. The reason is obvious, of course. A headphone (or any transducer) shouldn't add anything to the signal. It should only reproduce what's there.
4-Comfort. Although in order of absolute importance it seems (to me) to come in fourth, in truth you may as well put it in first place, because who the hell is going to wear uncomfortable headphones for extended listening sessions no matter how good they sound?
5-(For mixing ONLY in my production work, NOT for music reproduction for enjoyment) a slightly "analytical" quality, such as that provided by the Sony MDR-7506/V6. This slightly over-emphasized detail makes it much easier to hear tiny details in a mix. It does, however, slightly over-emphasize these details, which is why I prefer a more "laid-back" perspective (such as that provided by the Sennheiser HD-580 or HD-600) for simply listening to music for the fun of it. On an absolute scale I have no doubt that the HD-580 is more "accurate" than the MDR-7506. But I would never, NEVER use the HD-580 (or 600) as a monitoring device for critical recording/mixing!