I own a pair of DT880, but I use them rarely, because I don't like their sound signature. They're too bright for me due their excessive highs.
I mostly listen to Rock and Metal but I don't despise Jazz or Acoustic music.
I don't like much bass, espacially if it's bad qualaity bass. DT880's textured, fast bass was just right for me in terms of quantity too.
It's not a problem if it's a bit more, but not by a wide margin.
I don't like recessed mids either, especially if they're thin and emotionless.
I like smooth, inoffensive highs.
Budget is around 600 $.