The bicycle is an inverted pendulum (and is unstable when still) but similar to an inverted pendulum when it is stabilized through rocking back and forth the bicycle achieves this stability in motion.
The takeaway is that there are several factors which influence the stability of a bike. We know of certain designs which utilize one or more of these factors to achieve self-stability (and the conventional bike has all of these factors, hence why it tends to work so well), but we don't know the exact set of conditions on the combination of factors which would allow you to characterize which designs are stable versus unstable, without simply trying out any given design and simulating it.