Roger notes a piece on current research into how a bicycle works at UC Davis. The problem of bicycle stability is not completely understood, although real progress has been made in recent years - SciAm noted some of the work about a year ago
There are many issues other than stability - David Gordon Wilson of MIT , a gas turbine boffin who happens to be a cycling enthusiast, has written the best book I've seen - Bicycling Science. I recommend it if you have an engineering or science background. Not only does he talk about design and energy issues, but he goes into the history of the technology as well as a bit on other human powered tech - notably flight.
The video with the Davis piece concludes with the announcer mentioning that improvements in bicycle stability may increase bicycle use in the US. I seriously doubt that. Most people find bikes stable enough and don't worry about the issue. For the others there are a variety of trikes, some of which have advantages beyond stability. The real problem is much deeper and is societal. John Pucher of Rutgers is one of the best sources of information on the subject and his work is very readable. His lectures are also great if you have the chance to attend one. I've linked to a few of his papers - making cycling irresistible in the US and this survey paper are good stating point. There are many other papers, but this one that looks at Germany is useful.
One of John's major points is that integrating the bike into daily use is very complex, but a few societies have cracked the problem using, it turns out, very different approaches. One suspects that solutions in the US, if there are any, would be different, but that understanding what happened in parts of Europe is important.
One of his more interesting insights is that women are an indicator species for cycling.