These are in the DT:

WHEN A DOG SMILES - Mark Cohen
MY DOG'S BIGGER THAN YOUR DOG - Tom Paxton
ME AND LITTLE ANDY - Dolly Parton
OLD DOG TRAY
OLD BLUE
BESTEST FRIEND

So are these, but less directly relevant:

MR BOJANGLES (mainly the verse about "His dog and him travelled about. The dog up and died, ... but after 20 years he still grieved."


And on the less serious side:
HOW MUCH IS THAT HOUND DOG IN THE WINDOW
HOUND DOG SONG
not to be confused with:
HOUND DOG