To pick up on some of the songs mentioned above... I'm speaking here purely about the folk clubs in London I frequent:

Martin Said to his Man - I've heard this sung once or twice at the London folk clubs I frequent, over the last few years. It's not common, that's true, but it's not unheard of.

Manchester Rambler - this gets sung from time to time. A lot of the rambling/trespass songs are finding new popularity among 20somthings. It tallies with the trespass movement.

'Shoals of Herring' is heard quite a bit, mainly because of its inclusion in Inside Llewyn Davis.

Whaling songs and mining songs still get sung, mainly because they are predominantly about having to do dangerous, shitty jobs for not much pay - something a lot of young people can relate to. Hunting songs on the other hand don't get sung, though I think I might have heard some of those rare songs that are sung from the animal's point of view.

I never hear 'The Jew's Garden' anymore: people know its blood libel origins so nobody sings it.

'Fields of Athenry' still gets sung (never liked it myself but each to their own)