  The universal Spencer and de Rham complexes of sheaves over a smooth or
analytical manifold are well known to play a basic role in the theory of
$\mathcal{D}$-modules. In this article we consider a double complex of sheaves
generalizing both complexes for an arbitrary supermanifold, and we use it to
unify the notions of differential and integral forms on real, complex and
algebraic supermanifolds. The associated spectral sequences give the de Rham
complex of differential forms and the complex of integral forms at page one.
For real and complex supermanifolds both spectral sequences converge at page
two to the locally constant sheaf. We use this fact to show that the cohomology
of differential forms is isomorphic to the cohomology of integral forms, and
they both compute the de Rham cohomology of the reduced manifold. Furthermore,
we show that, in contrast with the case of ordinary complex manifolds, the
Hodge-to-de Rham (or Fr\"olicher) spectral sequence of supermanifolds with
K\"ahler reduced manifold does not converge in general at page one.
