Ambassador at Large
My most important feature
It's hard to even contemplate the variety of cameras out there for sale. Regardless of any other feature, size, zoom power, and so forth, the one most important feature for me is image stabilization (IS). One brand now calls it anti-shake.
If you are going to take hand held photos, it's your only chance to be sure of sharp images.
Optical IS tends to be better than digital IS. Some models offer both. For me, Optical IS is a minimum requirement.
DSLRs offer much better control of depth of field, and much better image 'depth.' They are generally better in low light, too. For that, you pay with a large and heavier camera. A DSLR is almost impossible to use while riding.
Another feature that I find important for shooting from a moving bike is a wider angle lens. Lots of cameras offer huge zoom ranges, but the widest angle setting is still fairly narrow. A wide angle lens (like 25-28 mm equivalent (to a 35 mm camera)) when added to IS will allow some fairly sharp shots while riding.
Even if you just shoot people on the ground, the wider angle lens allows you to get closer. This makes the flash much more effective. Most small point-and-shoot cameras have pretty wimpy flashes. They are not much beyond 5 feet away. With a wider angle lens, you can shoot the same portrait at 3 feet, and the flash is effective, even, and under control.
If you attempt to shoot video from a moving bike, you especially want IS and a <28mm equiv. lens. That's what makes the Hero camera so well liked.
Oh, and waterproof. If you simply hold the camera, and have a place like your tank bag to stow it, fine. But a solidly mounted camera is a rain magnet. Murphy lives. You don't want to have to pull over and stop if it starts to sprinkle, or worse.
SO, recapping MY opinion: DSLRs for high quality, control, image depth. Not for hand held shooting in motion on a bike. IS is a minimum requirement unless every shot uses a tripod or unipod. Buy a camera with a wider angle zoom lens, 28mm or less. Frosting on the cake: additional optical viewfinder, or an LCD screen you can read in direct sun. Slim and light for your shirt pocket. Battery life you can live with. A camera you can shoot without looking at it (while riding).
Oh, and widely standardized SDHC capacity, not non-standard memory cards like Sony memory sticks. Even Sony has abandoned them.
If I had my druthers, I'd carry a small, thin, wide angle, IS camera in my pocket.
I'd have a DSLR (optional) with 28mm - >80mm in my top case.
And extra batteries and memory cards for both.
Oh, and one more thing: Before you go to all the expense of a decent camera, and the trouble of carrying it with you, make sure there's someone (possibly more) that might, just by chance, want to see your photos. That someone may be you.
I quit carrying a camera unless I'm doing a photojournalism piece. Nobody wants to be bored by my photos. I'd rather make an effort to remember what gold Aspens looked like. YMMV.
Most of my pictures just use words.