  We develop a convergence theory of space-time discretizations for the linear,
2nd-order wave equation in polygonal domains $\Omega\subset\mathbb{R}^2$,
possibly occupied by piecewise homogeneous media with different propagation
speeds. Building on an unconditionally stable space-time DG formulation
developed in [Moiola, Perugia 2018], we (a) prove optimal convergence rates for
the space-time scheme with local isotropic corner mesh refinement on the
spatial domain, and (b) demonstrate numerically optimal convergence rates of a
suitable \emph{sparse} space-time version of the DG scheme. The latter scheme
is based on the so-called \emph{combination formula}, in conjunction with a
family of anisotropic space-time DG-discretizations. It results in
optimal-order convergent schemes, also in domains with corners, with a number
of degrees of freedom that scales essentially like the DG solution of one
stationary elliptic problem in $\Omega$ on the finest spatial grid. Numerical
experiments for both smooth and singular solutions support convergence rate
optimality on spatially refined meshes of the full and sparse space-time DG
schemes.
