The Yi Action Camera on Amazon is pretty good for the price. It costs more than $50, though, and you need the case to mount it securely on your bike. I haven't tried any of the other cheap cameras.

Helmet mount will add noticeable weight to your head and they look really dorky. Aiming a helmet camera is more complicated than aiming a handlebar camera and you may have to adjust the mount every time you put on your helmet. Cheap cameras do not have optical image stabilization, so mounting on your head will be more stable than mounting on your handlebars. Head mount also records what you are looking at instead of a fixed view.

I see people doing both styles, so there is no single best answer. Helmet mounts are more common with mountain bikers, mostly for the stability issue. Handlebar mounts are more popular with road bikers because it is simpler and you can lean forward as much as you want without affecting the camera's aim.