  We introduce a (bi)category $\mathfrak{Sing}$ whose objects can be
functorially assigned spaces of distributions and generalized functions. In
addition, these spaces of distributions and generalized functions possess
intrinsic notions of regularity and singularity analogous to usual Schwartz
distributions on manifolds. The objects in this category can be obtained from
smooth manifolds, noncommutative spaces, or Lie groupoids. An application of
these structures relates the longitudinal propagation of singularities for
pseudo-differential operators on a groupoid with propagation of singularities
on the base manifold.
