On a stand
When I hoist the bike on a stand and spin the rear wheel with all I've got on the highest gear, then sit back and watch it slow down gradually, there is a point where the bike + stand start to jitter considerably. The jittering subsides soon after when the wheel slows a bit more.
I suspect this is the result of resonance. (This experiment should be easy to confirm on your side, as bike wheels are always trued and never balanced.)
The combined bike+stand have a natural frequency, looking at the stand as a cantilever held by gravity to the ground.
Even a perfectly true bike wheel has some imbalance—from the valve, the hole, a speed sensor, cogs with odd tooth numbers, or the manufacturer's embossed rather than printed logo and inscriptions.
The moment of unusual jittering occurs while the wheel is slowing down and the frequency of its rotation matches bike+frame's natural frequency.
On the road
While riding I start to feel the onset of a vibration at specific speeds (64 kph on a previous bike; 55.7 kph currently). I don't know whether this vibration would subside afterwards because I treat this as my personal "maximum safe" speed.
My best current guess is that this vibration occurs when the natural frequency of bike + rider resonates with the frequency of the two wheels reaching a specific RPM.
As cyclists we are tuned to be concerned to the slightest new sound on a bike as a sign of mechanical trouble. Some of these (loose valve nut, which I prefer to keep but often omit to tighten adequately) are harmless. Many others are serious.
By extension I find the onset of a vibration at a (higher) speed worrisome. Should every cyclist feel a resonance frequency at a given speed?
Even if this is one of those "no need to worry about it" events, simply knowing that resonance is to be expected would at least make the event a less worrisome one.
Shimmies
The type of vibration discussed above is repeatable, at a given speed. It is subtle—worrisome only because of the speed it happens at.
I'm familiar with another, distinct, type of wheel vibration. On my regular routes there are specific obstacles (usually a 1cm concrete edge following tarmac) that cause a far more intense vibration, or shimmy. I understood the latter (discussed in many other questions here) to be the result of an intense contraction in one or a few spokes, which then resonate briefly until they are damped. These shimmies are quite intense. They happen at any speed, even low ones. They are also downright frightening the first time I'm surprised on unfamiliar terrain since I've then omitted to unload the saddle, and their intensity is much higher, perhaps due to the load.
