In Edinburgh:

Sandy Bell's Bar on Forrest Road-- for Irish/Scottish instrumental stuff most nights of the week.   The Royal Oak on Infirmary Street for singing-- mostly folk-- every night of the week. It looks like a dive (no, I take that back, it IS a dive), but there is occasionally an absolutely magical session in there. Don't be put off by appearances. In most pubs in Edinburgh, you can pick up a copy of The Gig Guide, which will list pretty much all of the live music in town for the month.