I will discuss the interplay between hypersurface singularities, their Milnor algebras, and classical invariant theory of homogeneous forms. In particular, I will prove that a contravariant that associates to a smooth homogeneous form the Macaulay inverse system of its Milnor algebra preserves GIT stability. I will discuss some applications of this result, for example to the direct sum decomposability of polynomials, and many related open problems.