In his fundamental work, Quillen developed the theory of the cotangent complex as a universal abelian derived invariant, and used it to define and study a canonical form of cohomology, encompassing many known cohomology theories. Additional cohomology theories, such as generalized cohomology of spaces and topological Andr\'e-Quillen cohomology, can be accommodated by considering a spectral version of the cotangent complex. Recent work of Lurie established a comprehensive $\infty$-categorical analogue of the cotangent complex formalism using stabilization of $\infty$-categories. In this paper we study the spectral cotangent complex while working in Quillen's model categorical setting. Our main result gives new and explicit computations of the cotangent complex and Quillen cohomology of enriched categories. For this we make essential use of previous work, which identifies the tangent categories of operadic algebras in unstable model categories. In particular, we present the cotangent complex of an $\infty$-category as a spectrum valued functor on its twisted arrow category, and consider the associated obstruction theory in some examples of interest.