There are lots of options if you are willing to make something or adapt something that is similar. "Mirror Mounts" come to mind, they are basicly brackets that clamp over any tubing under 1/2 inch usualy. Certain ones, like the ones I bought from center fire antenna, and from MFJ enterprises, will accept a pl-259 style adapter in the bottom, and have a 3/8ths inch threaded adapter on the top. Many antennas use the 3/8ths adapter, so its handy for the antenna experimenter, and you can buy antenna mounts with a steel whip in them. Replacement whips can be found in most welding supply stores, so you can cut them away to your favorite band without worry. Total cost of parts, usualy 10 to 20 bucks including a short run of coax, the bracket, and the antenna itself, experimenters beware, youll start putting antennas on everything! That craftsman riding mower could use a sky hook, how bout the mountain bike, then the air conditioner, then the tool box, then the quad.... whats next the cat!?!?!
If your sure of the exact band or bands, and want a really solid setup, NMO mount antennas can be had in mirror mounts, or you could get a drill through mount, and make a bracket yourself and bolt the bracket onto the bike, luggage rack, or even off a large bolt on the rear axle.
1/4 wavelength whip is probably the rout you will want to go for ease and preformance. The old 246 divided by the frequency will give you the approx size of the whip. For 154.000 MHz the whip would be roughly 1.5 feet, for 465 MHz it would be roughly .52 feet or half a foot. For CB at 27 Mhz figure the 108" whip.
One option I always liked for mountain biking was to build the scanner into a protective backpack (using a moble and sealed lead acid battery) and mount the 1/4 wave antenna onto the backpacks aluminum frame. This lets you take the radio off the bike with a great antenna, without unplugging or being tied to the bike. You also use your body as a shock absorber with that setup, and get the antenna up over your body for slightly better coverage.