Let $C$ be a curve of genus $\geqslant 2$.
Let $K_C$ be its canonical bundle.

Let $m$ be an integer.
We assume that a generic element in the linear system $|mK_C|$ is a simple divisor, i.e., a divisor without multiple point.
Let $S\subseteq|mK_C|$ be the set of divisors which are not simple.

Is $S$ always a hypersurface, i.e., without irreducible component of codimension $\geqslant 2$ ?
If it is the case, could we calculate the degree of $S$ in terms of $g$ and $m$ (at least for $m$ large enough) ?

How about higher dimensional varieties ?
More precisely, we consider a variety $X$. Let $S\subseteq|mK_X|$ be the set of singular divisors.

[remark for v2] I began by considering curves in v1. I am convinced that the answer is positive. Thanks to Jason Starr and abx.

Let $X$ be a complex projective variety.
Let $K_X$ be its canonical bundle.

Let $m$ be an integer.
We assume that a generic element in the linear system $|mK_X|$ is a smooth divisor.
Let $S\subseteq|mK_X|$ be the set of singular divisors.

Is $S$ always a hypersurface, i.e., without irreducible component of codimension $\geqslant 2$ ?
If it is the case, could we calculate the degree of $S$ ?