  We define a notion of connection in a fibre bundle that is compatible with a
singular foliation of the base. Fibre bundles equipped with such connections
are in plentiful supply, arising naturally for any Lie groupoid-equivariant
bundle, and simultaneously generalising regularly foliated bundles in the sense
of Kamber-Tondeur and singular foliations. We define hierarchies of
diffeological holonomy groupoids associated to such bundles, which arise from
the parallel transport of jet/germinal conservation laws. We show that the
groupoids associated in this manner to trivial singularly foliated bundles are
quotients of Androulidakis-Skandalis holonomy groupoids, which coincide with
Androulidakis-Skandalis holonomy groupoids in the regular case. Finally we
prove functoriality of all our constructions under appropriate morphisms.
