  We solve a number of questions pertaining to the dynamics of linear operators
on Hilbert spaces, sometimes by using Baire category arguments and sometimes by
constructing explicit examples. In particular, we prove the following results.
  - A typical hypercyclic operator is not topologically mixing, has no
eigenvalues and admits no non-trivial invariant measure, but is densely
distributionally chaotic.
  - A typical upper-triangular operator is ergodic in the Gaussian sense,
whereas a typical operator of the form "diagonal plus backward unilateral
weighted shift" is ergodic but has only countably many unimodular eigenvalues,
in particular, it is ergodic but not ergodic in the Gaussian sense.
  - There exist Hilbert space operators which are chaotic and $\mathcal
U$-frequently hypercyclic but not frequently hypercyclic, Hilbert space
operators which are chaotic and frequently hypercyclic but not ergodic, and
Hilbert space operators which are chaotic and topologically mixing but not
$\mathcal U$-frequently hypercyclic.
  We complement our results by investigating the descriptive complexity of some
natural classes of operators defined by dynamical properties.
