In these parts, kids in the late Fifties and Sixties said this:

See ya later, alligator.
After a while, crocodile.
Not so soon, you big baboon.

The last line wasn't said in a mean way, rather a joking, laughing way.

In our family, it was Puddin Tane. At least, that's what it sounded like to me.

See you around if you don't turn square.
That's another Fifties/Sixties I remember.

My parents also said, "Sleep tight, and don't let the bedbugs bite."

It's not a rhyme, but my dad often said this as we passed silos or circular structures: "Run into the roundhouse, dear! He'll never corner you there!" Someone told me this is from an old song, but he never sang it.

I didn't see this one:
I see London! I see France!
I see ...'s underpants!

Liar, liar, pants on fire!

Sticks and stones may break my bones, but names will never hurt me!

An apple a day keeps the doctor away.

Red skies at night, sailors delight.
Red sky in the morning, sailors take warning.

I've heard practically everyone in my family use the lines above at some time. I'm not sure where we got started on this. My dad was the only sailor. Maybe they read it in the Farmers Almanac. I really don't know.

Fill in the names:
... and ... sittin' in a tree -- K-I-S-S-I-N-G!
First comes love, then comes marriage,
Then comes ... with a baby carriage!

I think this also was a jumprope rhyme, but even today it's used when affection is observed. I recall Jack McFarland on "Will & Grace" starting to recite this one but getting bogged down on the spelling part.