  A periodic trajectory on a polygonal billiard table is stable if it persists
under any sufficiently small perturbation of the table. It is a standard result
that a periodic trajectory on an $n$-gon gives rise in a natural way to a
closed path on an $n$-punctured sphere, and that the trajectory is stable iff
this path is null-homologous. We present a novel proof of this result in the
language of covering spaces, which generalizes to characterize the stable
trajectories in neighborhoods of a polygon. Using this, we classify the stable
periodic trajectories near the 30-60-90 triangle, giving a new proof of a
result of Schwartz that no neighborhood of the triangle can be covered by a
finite union of orbit tiles. We also extend a result of Hooper and Schwartz
that the isosceles Veech triangles $V_n$ admit no periodic trajectories for
$n=2^m,m\ge 2$, and examine their conjecture that no neighborhood of $V_n$ can
be covered by finitely many orbit tiles.
