  Polygonal billiards constitute a special class of models. Though they have
zero Lyapunov exponent their classical and quantum properties are involved due
to scattering on singular vertices. It is demonstrated that in the
semiclassical limit multiple singular scattering on such vertices when optical
boundaries of many scatters overlap leads to vanishing of quantum wave
functions along straight lines built by these scatters. This phenomenon has an
especially important consequence for polygonal billiards where periodic orbits
(when they exist) form pencils of parallel rays restricted from the both sides
by singular vertices. Due to singular scattering on boundary vertices, waves
propagated inside periodic orbit pencils in the semiclassical limit tend to
zero along pencil boundaries thus forming weakly interacting quasi-modes.
Contrary to scars in chaotic systems the discussed quasi-modes in polygonal
billiards become almost exact for high-excited states and for brevity they are
designated as superscars. Many pictures of eigenfunctions for a triangular
billiard and a barrier billiard which have clear superscar structures are
presented in the paper. Special attention is given to the development of
quantitative methods of detecting and analysing such superscars. In particular,
it is demonstrated that the overlap between superscar waves associated with a
fixed periodic orbit and eigenfunctions of a barrier billiard is distributed
according to the Breit-Wigner distribution typical for weakly interacting
quasi-modes (or doorway states). For special sub-class of rational polygonal
billiards called Veech polygons where all periodic orbits can be calculated
analytically it is argued and checked numerically that their eigenfunctions are
fractal in the Fourier space.
