If you want a pair of headphones that will be good with all genres, it will be relatively flat in frequency response, or have a slight tilt in the bass. Essentially, there is little to no bass roll off that can be bumped a little. Mids and highs would be in line. Depending on listening preferences, the upper-mids/lower-highs may or may not be recessed (some people don't like this area of the spectrum flat as it causes thin vocals). Then you also have to account for the idea that modern songs are more compressed than the past rivals. This can create unrealistic boosts in the sound which I believe is why a slightly toned down treble is helpful as they will be more forgiving. So technically, not only is the person who listens important, the notion of time, when the song was released and how it was mastered, will impact it. So is there a signature that will suite everyone and everything? No. Is there a balance that will make just enough compromises on either end that will be enough for a single person, that's more likely. Does this all sound confusing? Yes, yes it is.
The thing is that a lot of this weighs in with the person listening to the headphones. It won't be Mshenay or myself that will need to know, it's Nirra that needs to know themselves. Part of the mission of the audiophile is to figure out what he/she likes so they can avoid what they know they don't like. There will be a nice set of headphones that works well with all genres, but then you go off and add yourself to the picture and that will change. Finding the perfect headphone for you can be a long journey.
As headphones get closer to being a jack-of-all-trades, they get quite expensive:
- NAD VISO HP50
- Bowers & Wilkins P7
- Westone W4
- Phonak PFE 1xx + green filters
The last in the list uses filters which allow you to customize the sound to your liking. It may be a good safe route.
Mshenay stated that the UE 6000 may fit your needs. If the 6000 is anything like the 9000 it's going to be a little dark for some things. If you turn on the internal electronics, you get a nice v-shaped signature that would be horrible for classical. It may work well for rock, pop, and rap though. Those three genres really only do need a slight v-shape to work well. Something like a Phiaton MS500 Bridge or V-Moda M-100 would work well with those 3 genres. In the lower price ranges, it's really tough to find a headphone with the good bass extension you'd need for something like rap or rock.