I would use cellulose or cotton insulation. Rock wool works well too. Look into quiet rock as well. Good luck.

Google "hat tracks" and "green glue". We used them for sound and insulation for a downstairs apartment and they work really well.

While expensive spray foam insulation is absolutely the best that I found for both thermal and sound!

One of the best products you can install is a sheet product called Quiet Brace. It has a sound reduction quotient of 26 which is a lot better than drywall. I have used it to build out recording studios. Ideally using metal studs for new walls you want to sound insulate would be optimum but may not apply to your situation. Denim insulation helps a little but the main things to reduce sound is density, and de coupling. The denser a surface is the more sound it will block. De coupling just means that two surfaces that conduct sound waves are not attached to one another...or they are cushioned from one another. If you could install denim between studs, a layer of 5/8 rock, a layer of quiet brace, then another layer of 5/8 rock glued together and screwed into studs...youd have optimum sound reduction