She sings silly songs by the seashore

It must be summer that's doing it to us, Therese. Building on your opus of jump rope rhymes, I've found myself thinking of the silly riding-in-the-car songs that we've sung with our kids, many of which go back to my own elementary school days. Don't tell anyone, but we still break out into these sometimes on a long drive or sitting out on our porch.

How about:

A peanut sat on the railroad track
His heart was all a-flutter.
Along came a big freight train.
Toot, toot. Peanut butter.

or,

Down by the bay
Where the watermelon grow
Back to my home
I dare not go.
For if I do
My mother will say
Did you ever see a Fish
Washing a dish
Down by the bay.
(What you see and what they're doing grow in ludicrousness as the song goes on.)

And here's a fun one:

Sing "Row Row Row Your Boat" quickly.
Start again, sing all but the last word.
Start again, omit the last two words, then the last three, and so on.

By the time you get to "Row row row your" "Row row row" "Row row" "Row", if the group is not convulsed with laughter you have more self control than we ever did.