I was born in 1956 and my sister in 1960. We grew up in the urban northeastern USA. Singing was more of a girls' thing back then. One endlessly-repeating song my sister used to sing was

Miss Mary Mack Mack Mack
All dressed in black black black
With silver buttons buttons buttons
All down her back back back
She asked her mother mother mother
For fifteen cents cents cents
To see the elephants elephants elephants
They jump so high high high
They reach the sky sky sky
And she never came back back back
And that was the end end end
Of Miss Mary Mack Mack Mack
All dressed in black black black (etc.)

The music for every line of this song was the same:
Tonic, second, third, fourth, fourth, fourth

She was also fond of singing a version of My Little Playmate:

My little playmate
Come out and play with me
Climb up my apple tree
Hee hee hee hee hee hee
Slide down my rainbow
Into my cellar door
And we'll be jolly friends
Forevermore forevermore

One song I can remember us boys singing at around age 11 or 12 went to the tune of John Brown's Body:

She waded in the water and she got her ankles wet (repeat 3 times)
But she didn't get her wet... yet

In each verse the water would rise another anatomical landmark up her leg until after she had gotten her thighs wet in the previous verse the concluding verse ran:

She waded in the water and she finally got it wet (repeat thrice)
Yes she finally got her bathing suit wet!

Back in the 80's a friend's elementary school aged daughter used to chant this:

CIN der RELL ah DRESSED in YELL ah
Went upstairs to kiss a fellah
Made a mistake and kissed a snake
How many stitches did it take?