Suppose $X$ is a locally convex space. Since the distributions $\mathcal{D}'\!(M)$ ($M$ a manifold) are a nuclear space, there is a canonical meaning to the topological tensor product $X\,\widehat{\otimes}\,\mathcal{D}'\!(M)$. Is there some way to define the pullbacks for such distributions (along an inclusion $N\subset M$ of a smooth submanifold)? For ordinary distributions, this is precisely when the wavefront set is disjoint from the conormal bundle of $N$. When $X$ is finite dimensional, there exists besides the wavefront set also the polarization set which gives more refined information on the singularities, but I have not found a comparable characterization of the existence of pullbacks, nor do I know what to do in the infinite-dimensional case (see e.g. this paper by Sahlmann and Verch, p. 3, for references on the polarization and wavefront sets of vector-valued distributions in the finite-dimensional case).
I believe this question to be very well motivated by QFT since we often want to integrate conserved currents over submanifolds to obtain conserved charges. However, these currents are actually unbounded-operator-valued distributions, so integrating them over submanifolds makes no sense a priori and one needs some sort of pullback. I have seen at least one workaround for this in the literature (in this excellent book), but it did not appeal directly to this kind of microlocal analysis, which strikes me as the most natural approach. (See also my other question on Math StackExchange.)
