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.