Suppose we have a projective flat non-smooth morphism of Noetherian schemes $g: X \rightarrow S$. My question regards when the sheaf of relative Kahler differentials $\Omega\_{X/S}$ is flat over $S$. In particular, I am wondering about the case where $S$ is the spectrum of a Dedekind domain, so we can just consider that case, if it makes things easier.

For example, if the geometric fibers are reduced curves with at most ordinary double points, then this morphism is a prestable curve and the sheaf $\Omega\_{X/S}$ is flat over $S$ (Knudsen, projective of moduli space of stable curves).

I'm wondering about higher-dimensional analogues? How about if all the geometric fibers are reduced surfaces (even hypersurfaces in $\mathbb{P}^3$) with at most ordinary double points? If this works, would any isolated hypersurface singularities work? If it is an arbitrary local complete intersection morphism?

Thanks!

Jordan