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Dunno if this has been posted yet but is interesting;
http://www.newstatesman.com/science/2013/08/we-still-don%E2%80%99t-really-know-how-bicycles-work
Seems like bikes are magic!
Clearly they are, and are generally propelled by awesome; and flapjack.
I thought that it was beacause of the head angle. If you imagine it being vertical, it would be very unstable and fall over but when that angle is put forwards towards the horizontal, when the bars are turned, the bottom of the wheel (where it is in contact with the ground is being put out to the side slightly. That adds stability and it is put back upright by the moment around the wheelbase as it turns due to the centripetal force cause by the change in direction of the bike. As the bike returns to the upright position, a slightly different part of the wheel is in contact with the ground in relation to the bottom of the fork which exerts a force on the bars which causes them to straighten. This is why a vertical fork would be so unstable, as the part of the wheel in contact with the ground is always the part of the wheel directly beneath the bottom of the fork. When the bike is not moving no force can be put on the wheel to straighten it up and so it falls over. This can theory can be extrapolated to explain the wobbling at slow speeds. That is my self derived theory. I hope it solves the "mystery". I will expect my nobel prize for bicycles shortly
What he said, but with paragraphs.
The thing is, it's not really a physics problem, it's a computing one - you balance by steering the bike back under your centre of gravity. The way your brain analyses pretty sparse data to create a feedback system to keep you upright, without you thinking about it, is pretty astounding.
There are some things that physics and maths can help with - with why some bikes are more stable than others, with how self-centering steering helps with the feedback system, with how the vertical movement as you steer can lead to "flop" or stability.
Over the years, people have tried to build "unrideable" bikes - ones with tiny wheels, ones with counter-rotating gyroscopes, ones with steering at the back or in the middle. Some take a while to learn to ride, but people get the hang of them eventually.
Just wait until they discover Di2 shifters, alien technology.