PhysLink.com tries to answer the question, and the author of the reply does not beat around the bush:
Yes, the gyroscopic forces, better known as the angular momentum, of the wheels on a bike allow us to ride a bike. … The reason that you stay up on a bike is that angular momentum, like regular momentum, must be conserved if no external torques act on the object. … When you are riding a bike forward, the right hand rule gives the direction of angular momentum to be to the left, perpendicular to the wheel. This direction does not want to change, therefore the wheel wants to stay upright and it makes the bike very ridable.
Wow. Flat out wrong in many ways and no mention of steering at all.
1. The David Jones Physics Today article from 1970, demonstrates that a bike is quite ridable if the gyroscopic forces are canceled.
2. Gravity acting on the center of mass and the ground reaction forces create a couple that acts as a large external torque. If there were no external torques, bikes wouldn’t tip over at all and there would be not need for a way to balance them
3. The direction of the angular momentum vector can easily be changed by the application of an external torque, such as the one described in the previous point.
The author even goes so far as to suggest that he has seen Jones’ article but comes to the opposite conclusion that Jones does.
However, when the extra wheel was spun backwards, the bike became almost impossible to ride because the vectors for angular momentum cancelled each other out. It was like trying to balance a bike that was not moving.
Instead, a main point of Jones’ article is how hard it was for him to make an unridable bike.