Cinderella, Cinderella, was common in my elementary school in Utah in the mid eighties. As was the Glory Glory Hallelujah. I can also add:

Joy to the World, the school burned down,
And all the teacher's died!
Except for the principal
We found him in the toilet bowl
With a rope around his neck . . .

And a rhyme for Chinese jump rope--

Ching, Chang, China Man
Chop Chop SUUU-EY!
Sitting in a china shop
Eating chop SUUU-EY!

Two other's I'm wondering if anyone else knows--

"What you say is what you are, You're a naked movie star" (I'm specifically interested in knowing if anyone knows if this was said before about 1980, or if it's known by anyone outside of Utah).

And one that I THINK I learned in Sacramento CA around 1989 that even my brother's and sister can't remember ever hearing before--I can't remember how it begins, but the part I do remember goes

"Have a peach, have a plum, have a stick of chewing gum,
and if you want another one, this is what you say--
A man, a man, a mandiego, sandiego
Hocus pocus dominocus
Brave Eagles!
Sitting on a trash can, beating on a tin can
Who can? We can! Nobody else can!
Sis, sis, sis boom bah!
Horses, Horses,
Rah! Rah! Rah!"

Also, the suffocation song--sung to the tune of Allouetta

Suffocation, super suffocation
Suffocation, the game we like to play

First you take a rubber hose,
Then you stuff it up your nose
Turn it on, then you're gone!
Oh, oh, oh, oh

Suffocation (repeat)

Next you take a pillocase,
Then you wrap it round your face
Go to bed, Wake up dead
Oh, oh, oh, oh

Suffocation (repeat)