By "regular" distributions I understand those Schwartz distributions that arise from locally-integrable functions. Are there ways of telling them apart from the non-regular ones? Does the set of those points at which (or around which) a distribution is (or is not) regular carry a name (something like the concepts of "support" and "singular support")?
Concretely, if $(M,g)$ is a Riemann manifold, $\Delta$ the Laplacian, $x \in M$ and $\delta_x$ the Dirac distribution at $x$, is $\mathrm{e}^{t \Delta} \delta_x$ regular on $\mathrm{R}\times M$?