If you want a handlebar mount for your bicycle or motorcycle, here’s an interesting solution: a hose clamp with a 1/4 – 20 flat head screw through it. This should work with point-and-shoot cameras, or with a GoPro using a Tripod Mount (Amazon).
Besides handlebars, the video after the “read more” lists a snowblower, stroller, cat, and car as mounting possibilities. I’m not sure how the “cat mount” works, but the others shouldn’t be too hard to figure out.
I would also think that a much larger hose clamp camera mount could be used. This would further expand the capabilities to things like larger PVC pipes, support beams, etc.
If you want to buy something purpose-built with more adjustment capability, this 360 degree mount (Amazon) looks interesting. Some users there reported problems with vibration/shock knocking it loose, so it might not be the best item for mountain bike use.