  Solutions to the wave equation in the exterior of a polyhedral domain or a
screen in $\mathbb{R}^3$ exhibit singular behavior from the edges and corners.
We present quasi-optimal $hp$-explicit estimates for the approximation of the
Dirichlet and Neumann traces of these solutions for uniform time steps and
(globally) quasi-uniform meshes on the boundary. The results are applied to an
$hp$-version of the time domain boundary element method. Numerical examples
confirm the theoretical results for the Dirichlet problem both for screens and
polyhedral domains.
