  A singular foliation is a partition of a manifold into leaves of perhaps
varying dimension. Stefan and Sussmann carried out fundamental work on singular
foliations in the 1970s. We survey their contributions, show how diffeological
objects and ideas arise naturally in this setting, and highlight some
consequences within diffeology. We then introduce a definition of transverse
equivalence of singular foliations, following Molino's definition for regular
foliations. We show that, whereas transverse equivalent singular foliations
always have diffeologically diffeomorphic leaf spaces, the converse holds only
for certain classes of singular foliations. We finish by showing that the basic
cohomology of a singular foliation is invariant under transverse equivalence.
