Murder ballads like "The House Carpenter", "The Twa Brithers", and "Oxford City", as well as bawdry like "The Little Ball of Yarn" and "Ten Nights Drunk" are sung by the tinkers but also by house-people. They also sing songs in the Gaelic, esp. love ballads like "The Red-Haired Man's Wife".
The only other songs I know about tinkers are "Robin Hood and the Tinker" - a very old (and very long) song, and "The Jolly Tinker", but that's exceedingly bawdy and not fit to repeat (until we've all had a coupla beers).