Why can a bicycle be ridden hands-free and stay upright riderless?

Partly settled. Some of it is understood, some is still argued about.

Kapitza's problems #19 and #68 ask why a bicycle can be steered hands-free and why the front fork is raked at an angle. For over a century, textbooks attributed riderless bicycle self-stability to gyroscopic precession of the spinning front wheel and/or 'caster' trail geometry. Kooijman et al. (2011, Science) built a bicycle with counter-rotating wheels to cancel gyroscopic angular momentum and negative trail to remove the caster effect, and it still balanced itself, proving neither classic explanation is necessary. A complete, simple, general account of exactly which combination of mass distribution and geometry is required for self-stability across all bicycle designs remains an active topic in dynamics research.

Generations of physics textbooks confidently explained a bicycle's self-balancing act with two mechanisms that turn out to be neither necessary nor sufficient, overturned by literally building the counterexample.

Explanations that circulated

• Front-wheel gyroscopic precession alone keeps a riderless bike upright (the 2011 Science paper built a bike with counter-rotating wheels that canceled gyroscopic spin and it still self-stabilized)
• Positive trail/caster effect alone is necessary for self-stability (the same experiment used negative trail and stability persisted)

References

• A Bicycle Can Be Self-Stable Without Gyroscopic or Caster Effects, J. D. G. Kooijman, J. P. Meijaard, J. M. Papadopoulos, A. Ruina, A. L. Schwab, Science, 2011
• Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review, J. P. Meijaard, J. M. Papadopoulos, A. Ruina, A. L. Schwab, Proceedings of the Royal Society A, 2007