  We discuss the two closely related, but different concepts of weak and almost
weak stability for the powers of a contraction on a separable Hilbert space.
Extending Halmos' and Rohlin's theorems in ergodic theory as a model, we show
that the set of all weakly stable contractions is of first category while the
set of all almost weakly stable contractions is of second category and is
residual. Analogous statements for unitary and isometric operators are also
proved.
