  This paper investigates the existence and properties of a Bernstein-Sato
functional equation in nonregular settings. In particular, we construct
$D$-modules in which such formal equations can be studied. The existence of the
Bernstein-Sato polynomial for a direct summand of a polynomial over a field is
proved in this context. It is observed that this polynomial can have zero as a
root, or even positive roots. Moreover, a theory of $V$-filtrations is
introduced for nonregular rings, and the existence of these objects is
established for what we call differentially extensible summands. This family of
rings includes toric, determinantal, and other invariant rings. This new theory
is applied to the study of multiplier ideals and Hodge ideals of singular
varieties. Finally, we extend known relations among the objects of interest in
the smooth case to the setting of singular direct summands of polynomial rings.
