Let $D$ and $S$ be two regular schemes and let $D$ be a divisor of $S$. Let $C \to S$ be a finite flat morphism, branched along $D$. Is $C$ regular as well?

The answer is yes when $\dim D=2$ and the variety upstairs (i.e. $C$ in your notation) is normal: see Bas Edixhoven, Robin de Jong, Jan Schepers, Covers of surfaces with fixed branch locus, Lemma 2.1. It seems plausible that this proof can be extended in any dimension $\geq 3$, although I did not check it carefully. 

