Foliations are well-behaved decompositions of a manifold into submanifolds with certain properties, called leaves. Here we focus on the more recent definition of singular foliations after I. Androulidakis and G. Skandalis, which carry more data than just a leaf decomposition, and we define singular Riemannian foliations (SRF) over this family.
Finally, we present a notion of equivalence relation deciding when two SRFs have the same transverse geometry, and we find an invariant through Poisson geometry. This is a joint work in progress with T. Strobl.