  We study regularity properties of solutions to operator equations on
patchwise smooth manifolds $\partial\Omega$ such as, e.g., boundaries of
polyhedral domains $\Omega \subset \mathbb{R}^3$. Using suitable biorthogonal
wavelet bases $\Psi$, we introduce a new class of Besov-type spaces
$B_{\Psi,q}^\alpha(L_p(\partial \Omega))$ of functions
$u\colon\partial\Omega\rightarrow\mathbb{C}$. Special attention is paid on the
rate of convergence for best $n$-term wavelet approximation to functions in
these scales since this determines the performance of adaptive numerical
schemes. We show embeddings of (weighted) Sobolev spaces on $\partial\Omega$
into $B_{\Psi,\tau}^\alpha(L_\tau(\partial \Omega))$, $1/\tau=\alpha/2 + 1/2$,
which lead us to regularity assertions for the equations under consideration.
Finally, we apply our results to a boundary integral equation of the second
kind which arises from the double layer ansatz for Dirichlet problems for
Laplace's equation in $\Omega$.
