Hound Dog (Bay at the Moon)
Little Brown Dog (the nonsense one)
Granny, Will Your Dog Bite?
Old Blind Dog
Pretty Little Dog (aka Susannah Gal or Western Country)
Fox and Hounds (aka Pretty Little Dog, but not the same tune as above)

There are also dogs featured in:

Darling Little Joe
Play 'Rocky Top' ("brand-new blue tick pup")
Stagolee/Stacker Lee/Stagger Lee (the narrator's bulldog)
'Fore-Day Creep (another bulldog)