This writeup was originally called Counting-out rhyme

A short, rhythmical, often nonsensical verse designed to choose someone or something from a group. Counting-out rhymes are a living tradition among children to this day, a part of their own oral literature.

Counting-out rhymes are usually used for games where one person is needed to do an unpopular task, such as being IT, or they are used in the opposite way - to select one by one all those who are not it. Rhythm and rhyme are more important than meaning, and the rhymes are fun to recite and tend to have a good sound. They are often accompanied by complicated and elaborate rules. No child doubts the fairness of a counting-out rhyme, or the seriousness by which it should be performed.

Here is a small fraction of all counting-out rhymes that the world is full of:

Eeny, meeny, miney, mo.
Catch a tiger by the toe.
If he hollers, let him go.
Eeny, meeny, miney, mo.
Often followed by
My mother told me you are not it you dirty old dishrag you.
My mother says to pick this very one.

Scout scout you're out.

My mother and your mother were at the store
Your mother punched my mother in the nose.
What colour was the blood?
(Name the colour and spell it out, then say the colour)
... was the color of the blood.

One, two, sky blue, all out except you.

Engine, engine, number nine
Going down Chicago line
If the train falls off the track
Do you want your money back
(Child picks yes or no.)
N-o spells no you don't get your money back or
Y-e-s spells yes and you shall have you money back

Two horses in one stable and one jumped out.

Inka Binka bottle of ink,
The cork fell off and you stink,
Not because you're dirty,
Not because you're clean,
Just because ya kissed a (boy or girl)
Behind a magazine and You are it.

One potato, two potato, three potato, four.
Five potato, six potato, seven potato, more.

Many thanks to purple_curtain for this one:

My operation,
How many people in the station?
(a number is named)
The one that lands on number (x)
Will surely not be it.