I seem to end up coming back to a very small set of favourites. My go-to one for the last couple of years has been 'Sea Song' by Robert Wyatt, which I guess is at least fitting in some way.
Then I've got a couple of indie songs of yesteryear that have been in my repertoire for ages: a few 'Storm In Heaven'-era Verve ones; one by Witness (anyone else remember them?); maybe a Kathryn Williams number or two. Occasionally I might do a Pink Floyd song (usually off 'Wish You Were Here'), but really I did those to death while washing in my student years.
How about the rest of you?