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?



regularity of finite flat branched covers
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*](http://arxiv.org/pdf/0807.0184v2.pdf), Lemma 2.1.


It seems plausible that this proof can be extended in any dimension $\geq 3$, although I did not check it carefully.


