I'm always looking for a headphone that reproduces music truthfully and expressively. If it cannot involve me in the music, it's not worth listening to. That is to say, it must have a well-balanced frequency range. Technically speaking, it cannot be harsh on top, and bass must be controlled, never boomy. My favorite part of hi-fi listening is the amazing details that can be heard, so it is important that one component of the sound not overpower the soundscape. It is a very nice thing to have an amazing soundstage, but when push comes to shove, I listen to headphones for the intimacy of the experience. If I want pinpoint accuracy, speakers will always be king, but I've never heard a stereo set-up that puts me so close to the musician as headphones. They sing into my ears, they pluck the strings on my eardrums, I hear every breath. It can be a very cathartic experience.
A perfect headphone would scale well with other components, as one upgrades, the aural pleasure is increased. If it complements the strengths of your components upstream, it's a beautiful thing. Unfortunately, a headphone that resolves incredible detail also exposes the flaws of the preceding parts of the setup. That is one of the reasons why this hobby is not for the faint of heart, one must be prepared to get an entire ensemble together.
Taking form factor into account is also important. Generally speaking, an ugly headphone will not sell (I'm looking at you, K1000s...), so a great product should be good-looking, especially if it is a portable pair of headphones. Size and shape is important for non-portable cans as well as portables. I do a lot of listening laying in bed, so it can be a problem if the ear-cups are too large, as they are uncomfortable against the pillow. I'd imagine that a fair number of other people listen in bed, too, so it is an important thing to consider. Weight and compression is another concern, I have a closed pair that is very heavy and compresses very hard, and they are very uncomfortable to wear for an extended period of time. That has really affected how much I listen to them, even though I like the sound, I can't listen for very long.
That said, there does not exist a headphone that is all things for all people. In this community, people need to recognize that what works for some folks won't necessarily please them. It depends on the music you listen to, what you value in sound reproduction, even your mood at the time of listening. When that's all said and done, you can just sit back and enjoy the music.