(This is a somewhat fuzzy question.)
I believe that most of the "standard" complexity classes that one comes across in complexity theory are countably infinite, because they are defined in terms of decision problems that can be solved by Turing machines with certain prescribed capabilities, and the set of Turing machines (on a given alphabet, etc.) is countable.
But certain complexity classes, like ALL, are uncountably infinite. Are any other "practically important" classes that come up on complexity theory uncountably infinite? What about in computability theory?