  Motivated by the study of H\"ormander's sums-of-squares operators and their
generalizations, we define the convolution algebra of transverse distributions
associated to a singular foliation. We prove that this algebra is represented
as continuous linear operators on the spaces of smooth functions and
generalized functions on the underlying manifold, and on the leaves and their
holonomy covers. This generalizes Schwartz kernel operators to singular
foliations. We also define the algebra of smoothing operators in this context
and prove that it is a two-sided ideal.
