  It is well known that, by the Reeb stability theorem, the leaf space of a
Riemannian foliation with compact leaves is an orbifold. We prove that, under
mild completeness conditions, the leaf space of a Killing Riemannian foliation
is a diffeological quasifold: as a diffeological space, it is locally modelled
by quotients of Cartesian space by countable groups acting affinely.
Furthermore, we prove that the holonomy groupoid of the foliation is, locally,
Morita equivalent to the action groupoid of a countable group acting affinely
on Cartesian space.
