What about "BINGO"? And Stephen Foster's "OLD DOG TRAY"? And "MY DOG'S BIGGER THAN YOUR DOG" (which is by either Tom Glazer or Tom Paxton, but not Tom Hood, and was later used in a Ken-L-Ration commercial)? And "OLD BLUE":

"I had an old dog

And his name was Blue,

And I bet you five dollars he's a good dog too.

Oh there's very few

Dogs like my Old Blue."

And let's not forget one that's not a folksong at all: "Ben", as performed by Michael Jackson.