I was primarily interested in the following question. Let $n\geq 3$, and let $X\subset \mathbb{P}^n$ be a degree $d$ hypersurface. Assume that its singularity locus $S$ (with reduced structure) is irreducible and smooth of dimension $k$. (As pointed out by @abx , this should not be a strict inclusion) Is it true that $$ \dim S^\vee \geq \dim X^\vee ? $$ The statement is true for quadratic hypersurfaces and hypersurfaces with isolated singularities. I am wondering if in general this holds.

Also, is there any criteria on when this inequality is strict ?

I was reading the book Discriminants, Resultants, and Multidimensional Determinants by Gelfand-Kapranov-Zelevinksy. In the first chapter they introduce the Katz dimension theorem, which computes the dimension of the dual varieties via local coordinates. But I found it difficult to use, since it's quite hard to write down the local coordinates. Is there any comments on the Katz dimension theorem, especially on how to use it ?