Casting my mind back, here are some I've never heard anyone sing before:

Buckskin Bag of Gold
Jeff in Petticoats
Rough on Rats
The Horse Wrangler (well-known, but never heard)
The Jolly Old Country Squire
The Footboy
My Dog and My Gun (the one by Bickerstaff)
Going Out a-Shooting (though the tune's well-known from King of the Cannibal Islands)
Comrades, Fill No Glass for Me (by Stephen Foster; okay, heard it once)

Then there's a bunch of old songs for which I couldn't find tunes, so I wrote my own; see my vanity thread.

And there's a slew of songs I've only heard one performer or group do (though they could be wildly popular in the UK and I'd never know).

Sometimes great songs are dug up out of obscurity for movies and documentaries, and thereby given a new chance at life. A few such songs that come to mind are:

He'd Have to Get Under - prominently featured in Horatio's Drive
The Girl in the Little Green Hat - used in Mrs. Henderson Presents
Simply Crazy (and) You Don't Love as I Do - sung by Madeline Kahn in Sherlock Holmes' Smarter, Younger Brother (or whatever the title is)

which reminds me of

You'd Be Surprised - sung by Madeline Kahn at an Irving Berlin tribute