Do shorn sheep regrow their wool?

Solution 1:

Sheep can regrow wool by eating a grass block (converting it to dirt). In addition, if you dye the sheep and then shear it, the wool grows back coloured. So this way you can generate flocks of rainbow sheep.

History: In early versions, after eating three grass blocks, they would regrow their wool(source). This disappeared (sheep would no longer eat grass), and then was added again in Minecraft 1.1.

Solution 2:

No.

Sheep don't eat grass anymore, and cannot regrow their wool.

Most of the time, you will not see any more sheared sheep because they have gotten out of your render range and "disappear."