Bizarrely enough, there is a pretty thriving jazz scene here. A lot of the upscale hotels have jazz bands, and during the winter the nostalgic tram down istiklal sometimes features a quartet. Other than that, good places to hear jazz are the Tunel district by the Galata tower, which features a LOT of jazz clubs with pretty legit pedigrees. The area also is something of mecca for musicians- it's full of little hostels and music stores, and you can always find surprise shows on the street out there. The college campuses, especially Bogazici generally have cool jazz or classical concerts going on every couple of weeks, too. Check Last.fm for tickets or the DKR bookstore on Istiklal for BILETIX ticket purchasing.
P.s. The guys who play in the Taksim metro from 3-7 every weekday are my favorite part of my commute. Totally worth the metro fees.