Depends on what you want from a Camera really.
But into the blue I second the Canon S90 or newer, better S95 version.
Square (nothing protruding when off), fair price and compact for its high degree of over-riding automatic functions. You might want the later to show speed (long shutter aperture) or freeze movement (short shutter aperture, fill-in flash). or when taking photos against sunsets etc..
It’s got a nice wide angle and useful tele. It’s got good lenses so that you can make decent pics without flash even in lowly light conditions.
I carry mine around in the back pocket of Endura Singletrack shorts and they are not too heavy.
For bike photography you might also want quick reaction times and a focus that is good in “following” the chosen object. The S90/S95 is good in this area, too. A good allrounder. But there will be better alternatives. Some more expensive, other maybe not so good in other areas.
There are smaller and cheaper cameras about. But they will probably have less functionality.
Ixus is a route if you want this.
In general I find Canons easy and intuitive to use.
(sorry for lack of technical photographic terms but I hope you get what I mean)