  We investigate regularizations of distributional sections of vector bundles
by means of nets of smooth sections that preserve the main regularity
properties of the original distributions (singular support, wavefront set,
Sobolev regularity). The underlying regularization mechanism is based on
functional calculus of elliptic operators with finite speed of propagation with
respect to a complete Riemannian metric. As an application we consider the
interplay between the wave equation on a Lorentzian manifold and corresponding
Riemannian regularizations, and under additional regularity assumptions we
derive bounds on the rate of convergence of their commutator. We also show that
the restriction to underlying space-like foliations behaves well with respect
to these regularizations.
