We study interactions between the categories of
D
\mathcal {D}
-modules on smooth and singular varieties. For a large class of singular varieties
Y
Y
, we use an extension of the Grothendieck-Sato formula to show that
D
Y
\mathcal {D}_Y
-modules are equivalent to stratifications on
Y
Y
, and as a consequence are unaffected by a class of homeomorphisms, the cuspidal quotients. In particular, when
Y
Y
has a smooth bijective normalization
X
X
, we obtain a Morita equivalence of
D
Y
\mathcal {D}_Y
and
D
X
\mathcal {D}_X
and a Kashiwara theorem for
D
Y
\mathcal {D}_Y
, thereby solving conjectures of Hart-Smith and Berest-Etingof-Ginzburg (generalizing results for complex curves and surfaces and rational Cherednik algebras). We also use this equivalence to enlarge the category of induced
D
\mathcal {D}
-modules on a smooth variety
X
X
by collecting induced
D
X
\mathcal {D}_X
-modules on varying cuspidal quotients. The resulting cusp-induced
D
X
\mathcal {D}_X
-modules possess both the good properties of induced
D
\mathcal {D}
-modules (in particular, a Riemann-Hilbert description) and, when
X
X
is a curve, a simple characterization as the generically torsion-free
D
X
\mathcal {D}_X
-modules.