(West London, early 1950's)

We had several rhymes for choosing who would be 'on' or 'it' before a game.(ie the 'chaser' or a rope turner for skipping etc)
We'd stand in a circle and everyone would present two clenched fists. The 'dipper' went round the circle, touching each fist. (Her own were touched using her chin!) When the rhyme stopped, that fist was put behind one's back. This went on until only one person with the final fist was left. They were then 'it'. This is what the dipper said:-

One potato, two potato, three potato, four.
Five potato, six potato, seven potato MORE!

or (this one was just pointing at the assembled candidates for a game)

Ip dip dip,
My little ship,
Sailing on the water,
Like a cup and saucer.
The one that comes to number three,
Surely will be he.
One. Two. THREE!

And a very racist one:-

Eenie meenie miny mo,
Catch a n*****by the toe.
If he hollers let him go.
Eenie meenie miny mo.

I also remember a rather boring ring game, usually for small children.
(One child in the middle, others walking round holding hands) This one was sung, not chanted.

The farmer's in his den
The farmer's in his den.
Ee aye ee aye.
The farmer's in his den.

This went on with:-
The farmer wants a wife (farmer chooses a wife, who joins him in the ring)
The wife wants a child (ditto)
The child wants a nurse (ditto)
The nurse wants a dog   (ditto)
We all pat the dog (everyone approaches the unfortunate child chosen as the dog and thumps him vigorously on the back!)