The Internet is always full of opinions so the same rule of thumb applies here - I'm sure the OP is sifting through and building his/her ideas with the input obtained in this thread.
Speaking of opinions, here are my thoughts from the backpacking arena (not biking, I know, but the constraints on backpacking can be more stringent than biking = no wheels to roll your load)
My favorite go-to stove is an MSR Simmerlite white gas stove. I've been backpacking for several years. The early trips I went on I didn't have a stove - people in my group brought the cooking gear and we spread it around (someone carries the stove, someone carries the fuel, someone carries the pot, etc, etc). That method worked great. The stove was a classic Whisperlite - and worked very well for our 4+ group outings. My first personal stove was a Snow Peak Gigapower canister stove. It works, but by biggest gripe with it is I can't get the flame to stay low enough for the lower-temp cooking/simmering that I like to do. The Simmerlite isn't a whole lot better, but at least I can set it and it stays where I put the setting (doesn't creep up in power on its own). I also have a lot of fueling options with the white gas stove that I don't have with the canisters - I can change the fuel bottle size and gain more fuel, or I can keep the small bottle and have some spare ounces stored for refilling if I got in a pinch. lots of options.
Two other stoves I've used are the MSR Pocket Rocket and Jetboil self-contained pot/stove system. Out of all the stoves the Jetboil is by far the most innovative and has real cool accessories (like the french press for coffee - the first time I had coffee on a backpacking trip came out of one).
For weight and simplicity - the canister stoves are hard to beat.
For cooking capacity and reliability - I'll put my trust in a white gas stove.
The original question at hand here is "what are some good safe stoves". I have knocked over my canister stove before. I've over-primed my white gas stove before = fuel leaking on the ground. Neither resulted in anything more than lost dinner or a quick flame up, but as someone already mentioned on here - what makes a stove safe is the user. Accidents can always happen, but if you are smart with your stove (as in understand it's limitations and how it operates) you will be a lot better off. I, for one, in the years I've been backpacking, have never heard of a limitation on stove use. In fact, quite the opposite. I have heard of times when fires are banned in state/national parks
except for those contained for the purpose of cooking (stoves).
What might be a better question is "what is a less-accident prone stove?" My first thought is stability. How stable is the set up when you have a pot on it? A single canister on the ground doesn't offer a whole lot of stability. There are some kits that have stabilizers that you can put on the bottom of the canisters to widen the footprint. These help a lot. My second thought is spill-ability. Liquid fuel is easy to spill. At the same time a canister has fuel under pressure so that could present a whole other set of challenges.
Last thought - use what you are comfortable with and trust. If you haven't used it more than a time or two don't take it on your trip. Cook on it a few times on short trips or even at home before you go out at all so you have time to acclimate to its use and characteristics. That goes for all necessary gear you take - make sure it is up to par and you are up to par with it before you put your trust in it and rely in it 100%.