Even among experts the effect of gyroscopic precession on bicycles is a controversial topic. At typical bike speeds, and with the weight of typical bike wheels, the effect is nominal.

The pioneering research in the field was done by David Jones, who published an article called " The Stability of the Bicycle" in Physics Today in 1970.
(See http://ist-socrates.berkeley.edu/~fa...onesBikeBW.pdf)

He did a series of experiments where he attempted to create unridable bicycles by cancelling out the forces believed to contribute to stability. In one of his experiments he cancelled out the gyroscopic forces by mounting a wheel of equivalent weight on the front wheel, rigged so it would turn opposite the front wheel. He found this modification had no impact on stability, and he was able to ride this bike no-hands.

What's interesting is that at higher speeds, gyroscopic forces become significant. I've definitely noticed that going down big hills the steering becomes a lot heavier, and gyroscopic forces are a significant contribution to the stability of motorcycles. What's even more interesting is the way that these forces work together so that a bike handles pretty much the same way at 40 mph as it does at 10 mph. One of the things that Jones points out is that a bicycle has to have a very specific geometry in order to balance, and that every bike built in the past 100+ years has had steering geometry within a very narrow range of angles.