There are loads of 'em! Quite a lot listed in the "Gig Guide", which you can pick up in most Edinburgh pubs/ theatres, etc. The Royal Oak in Infirmary Street has something on every night, as does Sandy Bell's in Forrest Row, then there's the Tass, the Waverley, the Antiquary, the Ensign Ewart for fairly Central Edinburgh, and the Harbour bar, Chain Pier and the Shore bar -all nearer the Forth - have less frequent sessions, and ? Musselburgh and Portobello.
Others further out may be only weekly or monthly, e.g.Penicuik, Haddington, Loanhead (do a search for their own folk club websites for details).
I usually go to the Grey Horse in Balerno (1st Tuesday of every month), Dean Tavern, Newtongrange (every Thursday - guest night first Thurs of month, otherwise a singaround), and Linlithgow, Black Bitch pub, 3rd Sunday of the month in the afternoon (from 2pm.)
Good hunting!