I got bored at work so I made a list of what instruments everyone would play if they did play anything... keep in mind it's not complete and might(probably will) disagree with other peoples headcanons, these are just mine!
Izaya- Violin, Harp, Cello, piano
Kururi- Piano
Mairu – Flute
Celty – Celtic fiddle, lyre
Shinra- Bagpipes
Shizuo- Electric Guitar, drums
Kasuka- Cello
Erika- guitar
Walker- Trumpet
Kadota- Guitar
Togusa- None
Roppi – Learned to play harp but gave it up because it was a waste of time
Tsukishima- Recorder
Tsugaru- Mandolin
Psyche- Guitar, sings
Delic- Piano, Saxophone
Hibiya- Piano, Violin
Mikado- Flute
Kida- The spoons
Anri- Clarinet