Cellular resolutions of monomial modules were introduced by Bayer and Sturmfels. In this paper the authors apply Forman's discrete Morse theory (in the version of Chari) to reduce cellular resolutions to minimal free resolutions. The method proves successful for generic and shellable monomial modules. The latter notion is introduced in the paper. It includes several classes of monomial ideals, among them the stable and the stable squarefree ones. The authors thus obtain minimal free resolutions that originally have been constructed in a more algebraic fashion, for example Lyubeznik's resolution of monomial ideals, the Scarf complex resolution of generic monomial ideals by Miller, Sturmfels and Yanagawa, the Eliahou-Kervaire resolution and its squarefree variants. For a certain class of affine monoid algebras the authors can also construct a minimal free resolution of the residue class field.