The Stacks project

Remark 111.50.2. The $k$th Fitting ideal of $\Omega _{X/S}$ is commonly used to define the singular scheme of the morphism $X \to S$ when $X$ has relative dimension $k$ over $S$. But as part (a) shows, you have to be careful doing this when your family does not have “constant” fibre dimension, e.g., when it is not flat. As part (b) shows, flatness doesn't guarantee it works either (and yes this is a flat family). In “good cases” – such as in (c) – for families of curves you expect the $0$-th Fitting ideal to be zero and the $1$st Fitting ideal to define (scheme-theoretically) the singular locus.