Hausdorff Morita equivalence of singular foliations.

We introduce a notion of equivalence for singular foliations—understood as suitable families of vector fields—that preserves their transverse geometry. Associated with every singular foliation, there is a holonomy groupoid, by the work of Androulidakis-Skandalis. We show that our notion of equivalence is compatible with this assignment, and as a consequence, we obtain several invariants. Further, we show that it unifies some of the notions of transverse equivalence for regular foliations that appeared in the 1980s. [ABSTRACT FROM AUTHOR]