Does one "drive" a motorcycle or "ride" it?

I would say ride/rode a motorcycle.
Unless you were making a deliberate point about a car alternative = "I don't drive a car I drive a motorbike"

Ride means, as reported by the NOAD and the OED, sit on and control a bicycle or motorcycle for recreation or as a means of transport; it also means sit on and control the movement of an animal, especially a horse.

Diana went to watch him ride his horse.

She rode a Harley Davidson across the U.S.

I would use "drive" to describe operating most machines or motorized vehicles. You can drive a car, bus, truck, motorcycle, tractor. "Ride" would be necessary for bicycles, horses, donkeys and very large dogs.

That being said, the term "cattle drive" is a good example of the word's usage aside from the riding/driving meaning.

I would say it depends on the context of what you are saying.

If you need to state that

there is something funny about the way he is riding the motorbike

it is different than saying

there is something funny about the way he is driving the motorbike

or

where riding indicates the activity of sitting on, and driving indicates the activity of pushing forwards.

So... were you sitting on the motorbike whilst it was taking you for a ride? or were you pushing that motorbike around like a slave of your will?

A bike is ridden. Please see Google NGrams Viewer for some empirical evidence.

I caution against using "drive motorcycle" as a search term because, while hugely prevalent, it refers not to the action but to the type of drive-train.

Additionally: a horse is ridden when atop it but driven when behind it (as with a carriage or a plough horse).

The terminology is left over from the days before motorized conveyances, where to "drive" horses or cattle was to induce them to move in a desired direction. People "rode" horses by sitting on them. People "drove" carriages by inducing the horses pulling them to move. People "rode in" carriages or "rode on" wagons when they weren't the driver.

So now,

• if something is a one-person mechanical conveyance you sit on, like a horse, you "ride" it.
• if something looks more like a carriage than a horse, you "drive" it.
• If something looks more like a carriage than a horse, and you are not controlling it, you "ride in" or "ride on" it (depending on whether you are inside or outside).

If elephants (which more than one person can ride) had been more common in England, we might "ride" rather than "drive" cars.