Kid's songs, along with bawdy songs and army/navy songs are probably the purest examples of the folk tradition extant. Botkin collected som, Tony Schwartz collected some, Pankake collected some. So did many others, and I'd be delighted if somebody posted some of them.

Some of my personal favorites:

Margarite, go wash your feet, the Board of Health is across the street.

I should worry, I should care, I should marry a millionaire;
He should die,I should cry, I should marry another guy.

I should worry, I should fret, I should marry a suffragette...