Use software antishake. If you upload to Youtube they will automatically offer to do some shake removal. VirtualDub also has some good and free antishake filters. Other editing software may have options as well. You will reduce your overall resolution by doing this, either that or black-bar the sides form time to time.

Honestly I wouldn't attach a camera with optical antishake such as a Canon powershot to a bike. I would think that it would wreck the mechanism pretty quickly.