I use an older Kodak ZX1. After trying several bicycle mounts from Ebay that didn't do anything to combat shake, and some of which broke from the bumps, and some desperate schemes to tie the camera on with rubber straps on a rubber bed...
---one day on impulse at Lidl I bought a GorillaPod. This is tripod with flexible legs and rubber balls along each of the legs. You're supposed to wrap the legs around whatever is handy, on a bike the tubes or the rack or whatever. (I use it on the rack, because it's purpose is to look backward and record drivers harassing me in the narrow lanes, in support of a self-defence plea when their SUVs suffer expensive accidents with my Abus Grnit 54X U-lock, which makes a good alternative 3-pound hammer.) The GorillaPod works for cutting out the shake. But it is neither small, nor tidy, though it is pretty light (by my standards anyway). In fact, it is a great big ugly thing. But it works by stabilising the picture so that number plates are readable and that is all that matters.
Andre Jute