Used on walks mainly.
Muffin the daw-w-w-w--w-wg / Muffin she goes with me when I explor-r-r-r-r-rr-r-e / I put on her leash / we go for a walk / she's MY dog, for sur-r-r-r-rr-r-r-re
Pretty sure this one is a mis-remembering of the Digger the Dog commercial from the 70s.
When she comes over
Hey-y-y therr-r-r-r-re muffintop / muffintop tennessee-e-e-e-e-e-e
Rocky Top-derived, one would think, with a weird nod to muffin tops of Urban Dictionary fame. I am not proud of my behavior.
General purpose, more a chant than a song
muffin puffin / muffin puffin / muffin puffin
It goes on like that with slight variations. You can probably pick out the pattern.