  Rational polygonal billiards are one of the key models among the larger class
of pseudo-integrable billiards. Their billiard flow may be lifted to the
geodesic flow on a translation surface. Whereas such classical billiards have
been much studied in the literature, the analogous quantum billiards have
received much less attention. This paper is concerned with a conjecture of
Bogomolny and Schmit who proposed in 2004 that the eigenfunctions of the
Laplacian on rational polygonal billiards ought to become localized along a
finite number of vectors in momentum space, as the eigenvalue tends to
infinity. For any given momentum vector $\xi_0\in\mathbb{S}^1$ we construct a
continuous family of quasimodes which gives rise to a semi-classical measure
whose projection on momentum space is supported on the orbit $D\xi_0$, where
$D$ denotes the dihedral group associated with the rational polygon.
