Thursday's the Bluegrass night at the Tass: think the others are more Trad/folk-based.
Newtongrange Folk Club have a singaround every Thursday night at the Dean Tavern from about 8.15: a bit out of town but on a bus route: visitors always welcome.
Almond Valley Folk Club will be having their monthly session at the Mid-Calder Inn on Tuesday 21st from 8pm: a bit difficult to get to without a car. (in Mid-Calder!) About 12 miles west of Edinburgh.
The Antiquary in Stockbridge has a session on Thursday evenings, but I think it's more instrumental than song.