  We consider the discrete shrinking target problem for Teichm\"uller geodesic
flow on the moduli space of abelian or quadratic differentials and prove that
the discrete geodesic trajectory of almost every differential will hit a
shrinking family of targets infinitely often provided the measures of the
targets are not summable. This result applies to any ergodic
$\mathrm{SL}(2,\mathbb{R})$--invariant measure and any nested family of
spherical targets. Under stronger conditions on the targets, we moreover prove
that almost every differential will eventually always hit the targets. As an
application, we obtain a logarithm law describing the rate at which generic
discrete trajectories accumulate on a given point in moduli space. These
results build on work of Kelmer and generalize theorems of Aimino, Nicol, and
Todd.
