Here's a children's singing game that I didn't learn as a child, but as a young adult as an acting warm-up exercise at an arts and music camp, and is one that I consider to be a cousin to the clapping games, since keeping up the rhythm (or bluffing in rhythm if you mess up) is crucial:

ROCK, ROCK (A singing circle game for as many people as you can round up)

Rock, rock, how you wander
From one hand into the other
From one hand into the other
Is it fair? Is it fair
To leave poor ("It's" name) standing there?

Instructions:
Everyone not chosen to be "it" stands (or sits cross-legged on the floor or ground) in a circle, as close together as possible. With "it" standing in the center of the circle. Those in the ring hold their left palm up, fingers slightly curled, so that anything in that hand is hidden from "it". Each person in the ring then mimes passing a small object from their own left hand into the left hand of the person on their right (so that your right hand is in the palm of your neighbor's left hand just as your other neighbor's right hand is in your left hand -- got it?)

The trick is that there actually is a small object being passed around, but because of the way everybody is in sync, and the way the left hand is curled slightly, it is very hard-to-impossible to see exactly where the rock (or whatever) is. And if you do catch a glimpse of it, it has moved to another hand almost as quickly. While the object is is being passed, and the chant being sung, the person in the middle must keep an eye out, and call out the name of whoever s/he thinks has the rock. If they cannot guess correctly after three tries they are out. The song stops, and whoever does have the rock is the new "it".

:::Sigh::: It's a lot more complicated to explain than to do.

As I said, I learned this game as part of an acting class, and I was so mesmerized by the rhythm we had going on that I completely forgot to keep track of where the rock was. All of a sudden, this small piece of broken asphalt "materialized" in my left hand. I was so startled that I just about dropped it, but managed to recover the fumble and pass it on before the person in the middle could see....

There is a Canadian version of this game, with slightly different rules, posted here.