Riderless bicycles (hand launched) tend to self-stabilise whilst in forward motion.
The phenomenon has been extensively studied at the Delft University of Technology (NL) in collaboration with Cornell University (US) (videos here)
The physics behind this stability is 'poorly understood'. Various theories have been put forward - including front-wheel spin angular momentum (gyroscopic), or 'trailing caster' effects - but these are disputed. Full details here
More history of the study of bicycle self-stability here
For a recent (2016) article, see The bicycle problem that nearly broke mathematics in Nature, 535(7612): 338-341.
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