Do riders generally hang onto the reins when they fall?

It's actually a bit of a bone of contention - and the answer is "It depends."

Holding onto the reins during a fall can potentially injure a horse's mouth and poll. It can also result in you getting tangled in the reins and dragged or pulling the horse on top of you.

On the other hand, letting go can result in a loose horse running onto a road or similar.

Most of us are trained to let go unless there's a good reason to keep hold.