Interesting to see what was going on in 2002, or even the 1970s: sorry I wasn't in Edinburgh then, so can't help Guest on that.
In the meantime, sessions have come and gone, and many have had to move house on an almost regular basis as new managements come in and decide they don't want folk music any more or can make more money out of the big screens and football supporters.
Jack has summarised most of what's on in Edinburgh at the moment, but there are other things out in west and East Lothian and Midlothian. And does the Guid Grack Club still meet in The Waverley Bar? The Canon's Gait has mainly Scots Music Group players on Weds nights, and the Slow Session people go to Leslie's bar in Causewayside. And the Hebrides still has a tunes/singing session on Mondays from 4-7pm, followed by another session with an accordionist. The Tass is now called No 1.
If visiting Edinburgh, pick up a free copy of The Gig List from city centre pubs, which tell you where it's at for all types of music.