Let's see... Picard listens to classical music, Cochrane listens to rock, Riker plays the trombone, Data plays the violin, and Voyager's Doctor performs operas. But do you think they still have pop, rap, techno, metal, and other today's genres there on Earth in the 23rd/24th century?
Well, 'classical' has been around for hundreds of years so I don't see why not. Music evolves so I imagine the genre may stay the same but the sound would be very different. What was thought of as heavy 30 years ago is very different to what constitutes heavy today. It's almost impossible to predict what music will be like in the 23rd century. As long as there's still guitar music humanity should be fine.
Rock music will live forever.... techno, however, will perish. I have determined that. I bet Cardassians listen to punk rock. Bajorans listen to emo pop.