To the chorus of Georgy Girl by The Seekers:

Milhouse, Mi-i-il-house Mi-i-i-i-i-i-il-house

I sing-spell their names to the tune of the "It’s raining, it's pouring" song while spelling them in ASL, then finish with their name and my made-up name sign for each one. Lily (cat) is L rotated to a finger gun, Milo (cat) is the hang 10 sign, Cooper (dog) gets C rocked back and forth a few times, and Zoe (dog) gets her name air-written in cursive (with a few extra 'e's to stretch it out).

Nobody in my house, including myself, speaks (signs?) ASL. I just watched way, way too much Switched at Birth way back when. 😂

Edited for clarity.

I'm a little teapot:

"I'm a little dogger short and sweet, here are my pawpaws here are my teeth."

There should be like 10 more but for sone reason I can't seem to remember them 😄

Dishwala - Counting Blue Cars:

Tell me all your thoughts on dog Cuz I really wanna feed her

I have a bunch more but this one came to mind first

Honky Tonk Cat

My partner comes up with the jingles, and his biggest hit is,

"Beat your butt,

Beat your butt,

Beat your butt like a butt slap slut!"

I want to marry him again.

I know you gotta potty.

I know you gotta potty.

I know you gotta potty,

Cause the poop is in your body!

You mean like this gem (YouTube link) from the early days of the interwebs?

[stands cat up and dances his paws to 'mr. Bluesky.]

Crinkle Crinkle little cat

So you know where your noms be at

Dreamies for you Docklesocks

Fishsticks for you Scullypops

Crinkle Crinkle little cat

So you know where your noms be at

I had a theme for my old dog, but it didn't have any words to it. It was just a tune I would hum or go "doo doo doo" with.

Though my mom's dog was a white toy poodle, so sometimes we would sing "Mary Had a Little Lamb" for her, except it was "[Dog's name] is a little lamb..."

Calling them "songs" is generous at best; they're more like jingles with a heaping dose of copyright infringement.

🎶 Who? Is? My little sweet potato? 🎶

/Lift her paw to make her look like she's answering a question in class/

🎶 I yam! 🎶

🎵"I said maybe! You're gonna be the pup that saved me! Because afterrrr allllll! YOU'RE MY URZA DOG!!!" 🎶

Belted out at about a thousand decibels to the tune of Wonder Wall by Oasis. It's considered a war crime in Los Angeles county.

Bean will, Bean will CHOMP YOU! (sing it)

Bean will, Bean will CHOMP YOU!

A friend is staying at our house fur the week, and we're watching their pups during the day. They sing:

Toe beans, toe beans, toe beans toe BEEEeans, you're such a fuzzy pup, toe beans

(Jolene, Dolly Parton)

I was going to say “what, people do that? Thats fucking silly” and then my wife reminded me that I literally have a song for my cat, Nyarlathotep, that I sing to him when I pick him up and for some reason my brain forgot about that or simply thought it didn’t count.

Anyway, it goes to the tune of “Don’t Bring Me Down” by ELO

NYAR Nyar nyar NYAR Nyar NYAR nyar Nyrar nyar nyar nyuh nyar NYAR NYAR NYAR NYAR

She's our Ada Puhtaaayta

She's our Ada Puhtaaayta

She's our Ada Puhtaaayta

We dug her up.

She comes from underground.

You little shit,
you little shit, 
why don't you shut up,
its 2am...

you little shit,
you little shit,
why don't you shut up
like your brother has?
its 3am!

you little shit,
you little shit,
why do you zoomies
at 4am?
you make your brother
go on a run
you tap your paws
so very loud
its 4am,
its 4am
why do you do this?

its 5am
its 5am,
why do you still meow?
we fed you once
your brother twice
and yet you still meow...

you little shit
you little shit
and so is your brother 
please let me sleep
please let me sleep
i am so bloody tired

Context: two siblings cats, recently adopted. They take turns to meow at night, both asking for food and i guess to check where we are. Once fed they go to sleep... for a bit and then the other one will meow... i have not had a full night's sleep in 3 months...

Firstborn:

She's a silly Sally, silly Sally, silly Sally

She's a sweetie kitty, sweetie kitty, sweetie kitty

(On repeat. Usually for long enough for her to have lost and then regained interest in whatever I was doing while singing.)

Wild child:

MISTAAAAAR... boycat...

MISTAAAAAR... boycat...

HE'S MYYYYY... baby...

HE'S MYYYYY... boycat...

(Usually when he enters the room. Still has mixed success in distracting him from choosing violence...)

Chloe the Cat Song

Cuz she’s a Chloe McChloe,
(She is a…good Chloe)
Chloe McChloe!
(Oh yeaaaaah!)

You Are My Sunshine, but I change 'sunshine' to 'Penshine'. Svefn-G-Englar by Sigor Rós is another song that I would always spontaneously sing to her.

reminds me of this video essay where they ask the question how different animals feel about human music

Whispers of Your Death by Counterparts is exactly that, albeit not an entire album.