We looked at a wide variety of applications for omnidirectional wheels, or Mecanum wheels, here. And we saw at least one guy claim to have installed them on his Toyota. But one applications we'd never thought of was to put one on a bicycle.
After all, what would the point of that be? For engineer and former toy designer James Bruton, curiosity is enough of an end in itself. Thus he hacked together this bike with a transverse-mounted omni-wheel up front, to see how it would ride. (If you want to skip the fabrication part, the riding starts around 11 minutes in.)