Well, I'm not THE expert(that would be my boyfriend the Edinburgh native..I'll get him to post as well)...but I was there last July and the two places I loved the most were

1)The Ensign Ewert (top of the Royal Mile right next to the castle) which is kind of 'touristy' but all three nights I went they had GREAT music.

2)WHISTLEBINKIES!!!! really nifty place, just off the Royal Mile, (dont ask me exactly where-close to #1 tho-cos we walked)...had more trad music early on in the evening until about 11pm, and then at midnight things got crazy-a band called "The Roods" came in after their regular gig and played some folk/punk sort of something that really was great (amazing fiddler in the band!)

Actually, there were several more places all in the same area with music..start at the Ensign Ewert and then walk and listen for strains of music from the next pub..follow your ears:-)!!