As a layman, I would like a pseudo-scientific explanation of why a
2-wheel vehicle, like a bicycle, (usually) stays upright when in
motion.

Thanks!

Hello.

I've located several scientific explanations of this phenomenon:

From NASA's web site:

"It is well known that balancing a bicycle standing still is almost
impossible, while on a rolling bike it is quite easy. Why?
Different principles are at work in each case. Suppose you sit on a
bike that stands still, and find it is leaning to the left. What do
you do? The natural tendency is to lean to the right, to
counterbalance the lean with your weight. But in moving the top of
your body to the right, by Newton's 3rd law you are actually pushing
the bike to lean more to the left. Maybe you should lean to the left
and push the bike back? It might work for a fraction of a second, but
now you are really out of balance. No way!
On a rolling bike, balance is kept by a completely different
mechanism. By slightly turning the handlebars right or left, you
impart some of the rotation of the front wheel ("angular momentum") to
rotate the bike around its long axis, the direction in which it rolls.
That way the rider can counteract any tendency of the bike to topple
to one side or the other, without getting into the vicious circle of
action and reaction."
source: NASA.gov (18a) Newton's 3rd Law
http://www-istp.gsfc.nasa.gov/stargaze/Snewton3.htm 


From the Straightdope.com:

"Because modern bicycles are equipped with a pair of  gyroscopic
stabilization devices that require the motion of the bike in order to
operate. These devices are known as "wheels."
What is a gyroscope and how does its stabilizing power work? A
gyroscope is just something spinning. A spinning object has angular
momentum, whose magnitude is dependent on the speed of rotation, the
mass of the object, and the distribution of that mass with respect to
the axis of rotation. Angular momentum, like its homely cousin linear
momentum, is conserved. For our purposes this means that once a
gyroscope gets lined up in a certain way, it wants to stay lined up.
That, in short, is how gyroscopic stabilization works."
http://www.straightdope.com/mailbag/mangularmo.html

Also see: 

HOW THINGS WORK: Bicycles, hosted by virginia.edu
http://howthingswork.virginia.edu/bicycles.html

"Ask A Scientist: Bicycles," hosted by anl.go
http://newton.dep.anl.gov/askasci/phy00/phy00255.htm


search strategy: 
"balancing a bicycle", balance bicycle
"in motion", easier, "standing still"

I hope this helps.

Perfect! Just what I wanted.

And many thanks also to PinkFeud.

There's some useful information on this page (starting with the
heading "Balancing and Steering"):

Exploratorium
http://www.exploratorium.edu/cycling/brakes2.html

I didn't read the links but if the answer is saying that it is a
gyroscopic effect that makes a bike stable, that is 100% wrong!!! 
Actually, not 100%, but about 97%.  Only about 3% of the stability of
a bike comes from the gyroscopic effect of the wheels.  The effect is
actually much more straightforward than that.

It turns out that a bike in motion moves towards the direction of
lean.  (We all know this right?  You don't actually turn the handle
bars once your are moving, you just lean into the turn.)  The reason
for this is that the curved tire rotates along a straight line at the
tire ground interface only if it is totally perpendicular to the
ground.  If it is tilted one way or another it traces out an arc.

That tendency of the bike to follow the lean is a powerful stabilizing
force.  If provides a feedback loop that tends to make things very
stable.  What happens is essentially this.  A lean is initiated, the
bike's contact patch moves in the direction of the lean, a moment is
generated that counteracts the lean, the lean is reduced.

Some other points if you still question this.  If the gyroscopic
effect were large, the front wheel on bikes would turn by a large
amount when you leaned.  It doesn't.  If you ride a bike, and stand up
in the saddle, you can actually move the bike to and fro quite easily.
 This is because you are decoupling the large mass (you) from the
contact patch and therefore reducing the moment.  The bike can move
left and right, and with it the patch, with little resistance.  This
would not occur if angular momentum played a large role.  Finally, if
the angular momentum played a large role, wheel diameter would have a
very large effect.  It doesn't,  kids can ride bikes on their 16"
wheels just as easily as adults on their 27"  wheels.


Cheers.