I have used several different basic point-and-shoot digital cameras to take ride videos with good results. The only camera that didn't work for this purpose was a Kodak Z1285. It took great video for short rides, but the first attempt at a long ride and it broke. Probably couldn't handle the vibration.
I've used two different mounting methods that both work well. First was a plastic clamp meant to hold a reflector on bicycle handlebars. Camera was mounted to a screw that fit the tripod hole on the bottom and the screw was what held the clamp in place. The second method was a longer screw through the left side turn signal clamp and screwed into the tripod hole.
That's a small section of rubber fuel line between the camera and turn signal housing to avoid buzzing.