[Trapper]

From Rogers Yahoo Answer Man: (great weekly column)
http://yahoo.rogers.com/yahoo/answerman/index.jsp

Originally Posted by The Answer Man

The underlying reason why bikes in motion are more "stable" than stationary bikes is not that they are more rigidly "upright" or more resistant to the forces that would make them fall. Rather, it's because the tipping process is drawn out and delayed, so there's more time for you to recover. When a stationary bike begins to tip to one side, the next step is almost inevitably a topple, or a foot placed on the ground. But when a moving bike begins to tip to one side, the front wheel is first naturally inclined to turn in that direction before simply falling over. This gives you extra time to regain "uprightness" by leaning to the other side and/or turning the wheel back in the other direction before the bike falls.

One of the common reasons given for this is that your wheels function as gyroscopes; when they're in motion, they want to stay in motion. Or more precisely, they want to maintain their angular momentum. When you tip a spinning wheel on its side, the conservation of angular momentum requires another motion to counter it. (You can see this if you hold a spinning bicycle wheel in your hands and try to tilt it. The wheel will want to twist around in your hands.) The natural reaction is for the wheel to turn in the direction of the tilt: if some force causes the bike to tilt to the left, the wheel turns left.

This gyroscope effect does have an impact on balance, but it turns out it's a pretty minor one. In fact, experiments have been done on bicycles equipped with extra gyroscope-cancelling wheels, and they're still very ride-able, which suggests another force is at work.

The real reason you stay balanced is something called "trail" that occurs because of the angle of your steering axis. Unless you have a really crappy bike, you'll notice that the steering axis is slightly tilted; your steering fork isn't vertical, and it doesn't make a right angle with the ground. The place where the tire meets the ground (directly below the wheel's axle) is slightly behind the steering axis (draw a line from the handlebars down through the fork and to the ground). This "negative trail" leads to the same result mentioned above. The physics are a little complicated (please see the links below), but when the bike tilts to the left, its natural inclination is to turn to the left as well, maintaining temporary stability until you can readjust it to its original equilibrium.