They usually involve rolling contact, I love rolling contact, from childhood on:
-toy cars, scooter, roller skates, skate board, bicycle, moped, car, etc.

Bicycle stability see: http://bicycle.tudelft.nl/yellowbicycle/index.htm

Bicycle: highly unstable at low speed but at moderate speed stable:
Yellowbike1.wmv,
even self-stable
Yellowbike2.wmv!

A simple model (4 rigid bodies with constrains) shows identical behaviour:
bike3v440.avi.WMV

Were does the energy from the lateral motions go? Does it damp out? Where is the damping?
Yes, from the lateral motions to the forward motion, it speeds up a little.
So you could also propel a bicycle by exciting the lateral motions?
BikeAsmeGood.wmv
And a again a model to predict this:
BikeMoviePro2.mov.WMV

But you can also stabilize the unstable motions like the motorcycle team Ghostrider from Berkeley CA in the DARPA challenge to cross the Mojava dessert (price 1 Million USDollars) :
slow.wmv
or even balance at zero forward speed:
balancing.wmv
http://journalism.berkeley.edu/projects/mm/zack/

Or with LEGO Mindstorms:
http://bicycle.tudelft.nl/schwab/Bicycle/index.htm#May_2010

Who needs TWO wheels? Just use ONE:
dvd-to-mpeg.mpg
Charles F. Taylor's passionate hobby from 1939 on, was the development of several working prototypes of a one-wheeled vehicle. http://www.me.berkeley.edu/one_wheel_vehicle/