  We study the descriptive complexity of sets of points defined by placing
restrictions on statistical behaviour of their orbits in dynamical systems on
Polish spaces. A particular examples of such sets are the set of generic points
of a $T$-invariant Borel probability measure, but we also consider much more
general sets (for example, $\alpha$-Birkhoff regular sets and the irregular set
appearing in multifractal analysis of ergodic averages of a continuous
real-valued function). We show that many of these sets are Borel. In fact, all
these sets are Borel when we assume that our space is compact. We provide
examples of these sets being non-Borel, properly placed at the first level of
the projective hierarchy (they are complete analytic or co-analytic). This
proves that the compactness assumption is in some cases necessary to obtain
Borelness. When these sets are Borel, we use the Borel hierarchy to measure
their descriptive complexity. We show that the sets of interest are located at
most at the third level of the hierarchy. We also use a modified version of the
specification property to show that for many dynamical systems these sets are
properly located at the third level. To demonstrate that the specification
property is a sufficient, but not necessary condition for maximal descriptive
complexity of a set of generic points, we provide an example of a compact
minimal system with an invariant measure whose set of generic points is
$\Pi^0_3$-complete.
