My dad used to sing a song that Dave Barry had in his greatest hits book:
Adventure Dog!
Adventure Dog!
Kinda big, kinda strong
Stupid as a log.
That was our dog (RIP sweet Snuffles). The essay that Dave Barry had that in also never fails to crack me up.
Susan
Does a cat song count?
All my wife does is bouce the kitty in her arms, while singing, “You are so pretty,” over and over again to the tune of “Go Tell Aunty Rhody.”
I swear I’m going to tape her one of these days, just for the blackmail potential.
I do not do this much with the dog I have now, but as a kid in the 80’s I would do this a lot with the dog of my childhood. (RIP, Jessie. She was an Irish Setter.)
First, I would hold her by the snout (she didn’t mind) and sing into her nose. To me the nose looked a little like a microphone, so I would say, “Microphone Dog”, a few times, just to get the soundcheck, a course. Then, to the tune of the theme song from Mork and Mindy:
Pu-ppy, dog.
Pup Pup Pup Pu-ppy, dog.
Pup Pup Pup Puuu-uh-ah-uh-oh-ah-ah oh, oh oh, ah
puppy-dog, dog, dog.
hard to sound out that last bit, but maybe you get the idea.
(repeat)
i miss her.
I tend to make up limericks, so they’re not really songs, but…
There once was a doggy named Bella
She kind of looked like a fella’
She’d lie on the grass
And out from her ass
Would come the most horrible smella’
I frequently substitute their names for names in songs - “Peggy And Her Gallant Soldier” becomes “Penny And…,” and “Mallie Lee” becomes “Penny Lee.”
We sing “Mr. Mojo Rising” to the Mojo-dog. And we sing Mojo Nixon songs to him, since that’s who he was named after.
Hmmm…I can’t think of anything specific that we sing to Binkley.
My dad just makes up lyrics - I’ll never forget catching him singing to our Dachshund, to the tune of “The Battle Hymn of the Republic:” “I see the little Doxie and she’s sitting on the chair…”
He didn’t know anyone was home at the time!
Valen (nicknamed Duff, or Duffman) gets “DUFFMAN…” sung to the theme of Batman on a regular basis.
and to the Arrogant Worms song Billy the Theme Park Shark gets recycled as: “Valen, the theme park putz… performing tricks for tourists that have paid five bucks a head… Valen the theme park putz… he looks just like a killer but he’s too well fed…”
(That last one gets used for all dogs: Spanky, and Zap (sung as Zapper))
Zap gets the “You are my puppy, my only puppy, you make me happy when skies are greeeeey…”
and "If you’re a puppy and you know it shake a paw… "
We also sing variations on “Puppies in the niiiiight, exchanging putzes… lovers at first siiiight… feeling my nutses…”
Yeah.
We’re weird.
Wufs from the NoFair household…
Elly
I don’t sing to Brutus,but I talk to him,and he understands most
of what I’m saying.If I could find a woman with his qualities I
might consider remarring.
FrankoAuggie, The Cutest Dog on the Planet ™, regularly gets to hear the following, first sung by Yosemite Sam:
Oh, I can’t get along little doggie,
Oh, I can’t even get one that’s small,
Oh, I can’t get along little doggie,
Oh, I can’t get a doggie at all.
I also sing “Auggifer, Doggifer” to the tune of “Jennifer, Juniper”.
I love these songs!
I have many, but the general favorite “Going Out in the Morning” song (sung to the tune of “I Sing The Body Electric”) goes:
*I sing the puppy electric
I celebrate the dog yet to come
I toast when puppies go potty
and nuggets come out of their BUUUUUMS . . . *
They pretty much ignore me and bolt for the back door in the middle of the first line.
My dog is also Brutus 
My estranged husband’s father used to sing (not sure of the title to the original tune):
She is the dog, our puppy dog
She is the dog we have around the house
She is the dog who never caught a mouse…
I think there was more but I don’t know it all.
I try to be as understanding as possible. Trust me, though, you wouldn’t want to marry someone like me. Gun fetishes, picky eater, the list goes on.
Only on Tuesdays and Sunday afternoons!