The Lipman-Zariski conjecture asserts that a complex variety with a locally free tangent sheaf is smooth and has been proved for several special cases. In the article under review, the author proposes and studies a generalized version of the Lipman-Zariski conjecture. Let \((x\in X)\) be an \(n\)-dimensional complex singularity and let \(\Omega_X^{[p]}\) denote the reflexive hull of the sheaf \(\Omega_X^p\) of differential \(p\)-forms. Then the problem is whether the freeness of \(\Omega_X^{[p]}\) implies the smoothness of \((x\in X)\) for some \(p\). The author gives an example of negative answer with \(p=2\) for terminal singularities of dimension three and four, and proves that if \(p=n-1\), there are only finitely many log canonical counterexamples in each dimension, and all of these are isolated and terminal. Applying this result the author also shows that if \(X\) is a projective klt variety such that \(\Omega^p\) on its smooth locus is flat, then \(X\) is a quotient of an abelian variety. On the other hand, an affirmative answer for any \(1\leq p \leq n-1\) is obtained if \((x\in X)\) is a hypersurface singularity with singular locus of codimension at least three. The proof is based on the study of the torsion and cotorsion of the sheaves \(\Omega_X^p\) on a hypersurface in terms of a Koszul complex. As a corollary, the author obtains that for a normal hypersurface singularity, the torsion in degree \(p\) is isomorphic to the cotorsion in degree \(p-1\) via the residue map.