I just learned that the two main effects that people attribute to bicycle stability (gyroscopic torque and steering trail) are not completely necessary for a stable bike.
Check out the Science Friday and PDF links in the article below. I think the article overstates the case, though. While you can make a bike that doesn’t tip over without those effects, it’s likely not one you’d want to ride. In the SciFri video, the non-gyro non-trailing bike look barely stable to my eye — a nice proof of concept, but hardly a model for a new class of bike.
More detail: http://bicycle.tudelft.nl/stablebicycle/