It is quite difficult to balance in a stopped 2-wheeler but quite easy, once u hv got used to it, when the 2-wheeler is moving. Why's that ?

U need to think wrt the mechanics of the wheel n not anythin else...

This happens because of law of conservation of angular momentum.

The angular momentum is given by right hand law. A moving cycle will possesses angular momentum , to make it zero ,there has to be some external torque acting on it. This torque is exerted by the frictional force. So it takes some time for the cycle to stop and become off balance.

HOPE THAT HELPS.

A spinning top also stays balanced because of the same reason.

HOPE THAT HELPS.

I think strictly speaking there are angular momentums in three mutually perpendicular directions when we ride the bicycle,

Because of inclination(swaying) of bicycle to left and right there will be an angular momentum  in a horizontal axis. Now we know we can never keep the byicycle handle straight. Actualyy the handle always oscillates between left and right even if we traverse straight. That is why moment the handle is locked or stuck we will fall down. The front wheel path will never be a straight line. It will be a zig zag curve.

So in a bicycle there are three dimensional angular momentums involved,  one due to rotation of wheels(X-axis) ,one due to handle turning(Z-axis) and one due to tilting of the bicycle(Y-axis). Conservation of all these momentums contribute towards balancing. Absence of any one can spoil the balance.