How many times per second does your eyeball move?

There is a typical eye movement called "cicadic movement". It is a tiny vibrating movement. It happens so fast that it appears you are looking at a whole object, but your eyes are moving 70 to 100 times per second all over.

When there is tension in the eye, the cicadic movement starts to slow down, and your eyes try to see too much at once, so that your eyes aren't dancing any more. If you look at a young child or an adult with very good eyes, their eyes appear to sparkle.

People who are sick have dull and listless eyes because they are not moving like that any more. Our eyes are designed to move, and tension slows them down and we fall into staring.

http://www.consumerhealth.org/articles/display.cfm?ID=19990303181620

Of additional interest:

The dynamics of saccadic eye motion give insight into the complexity of the mechanism that controls the motion of the eye. The saccade is the fastest movement of an external part of the human body. The peak angular speed of the eye during a saccade reaches up to 1000 degrees per second. Saccades last from about 20 to 200 milliseconds.

In addition to the kind of saccades described above, the human eye is in a constant state of vibration, oscillating back and forth at a rate of about 60 Hz. These microsaccades are tiny movements, roughly 20 arcseconds in excursion and are completely imperceptible under normal circumstances. They serve to refresh the image being cast onto the rod cells and cone cells at the back of the eye.

In exploring the visual scene we make about three saccadic eye movements per second.

https://en.wikipedia.org/wiki/Saccade#Velocity_and_duration

Also see:

https://en.wikipedia.org/wiki/Eye_movement

http://www.medscape.com/medline/abstract/16226583