Is it just inertia that keeps morotcycles and bikes from tipping over? I’ve wondered how–granting balanced loads may be involved–they tend to stay upright when the conveyance is moving…
Part of it is the angular momentum of the spinning wheels resists sideways motion (perpendicular to the direction of spin) that would result in tippage. This is why it’s much easier to balance a moving bicycle than a stationary one.
There’s a very important and often-ignored aspect to the design of bikes - the forward-facing front forks. These mean that whenever the moving bike leans to one side, it automatically steers itself into the lean, making the adjustments necessary by the rider to be immediate, and therefore within the abilities of the average human. A bike with a vertical front fork would be unridable.
Bicycle stability is complicated. Most sources (such as this one) say that the angular momentum of the wheels contributes little to stability.
The most important aspect is trail (as GorillaMan indicates). The link above explains this fairly well. Googling “bicycle stability” produces all sorts of links.
There’s been threads on this before, they seem to turn into a shambles quite quickly.
Anyway, a lot of it, on a bicycle at least, is simply the rider continually steering to keep the wheels under the centre of gravity (unless turning/leaning). The faster you go, the smaller the movements required to keep balanced and so it’s easier. You can balance a bike at a stand-still but you need to be prepared to make big positive movements of the handlebars and front wheel as well as changing your body position.
Ther’e a staffreport on this which more or less gets it right: Why is it easier to balance on a moving bike than a non-moving one (revisited)?
Ooh, I’m a cite 
My moment of glory…