I have read that if a variety $X$ has KLT singularities, then it has quotient singularities in codimension 2. Do you know a proof (or where can I find a proof) of this?
(For some reason first I thought that the question was only in dimension $2$...) The essence of this statement (in dimension $2$) consists of two facts that are useful anyway:
For the general case see Prop. 9.3 in Greb, Kebekus, Kovács, Peternell: Differential forms on log canonical spaces, Publ. Math. Inst. Hautes Études Sci. No. 114 (2011), 87–169. A heuristic way of seeing why this should be true is the following: Take a klt singularity and localize at a codimension $2$ point. You get a klt surface which is a quotient singularity by the above argument. The problem with this approach is that the residue field at the codimension $2$ point is not algebraically closed so you would either have to pass to the algebraic closure or analyze singularities over nonalgebraically closed fields and then once you have the localized quotient singularity description prove that it comes from one in the original situation. All of this is possible to do, but not entirely selfevident. The proof referenced above proceeds in a different way. 


I guess you are considering normal singularities. In this case you can probably use [KollarMori, Birational geometry of algebraic varieties, Corollary 5.21 p. 161]. For the reader's covenience, let me write the complete statement. Proposition. Let $x \in X$ be a germ of a normal singularity. (1) $x \in X$ is Kawamata Log Terminal if and only if it is a cyclic quotient of an index $1$ canonical singularity $0 \in Y$ by an action which is fixed point free in codimension $1$ (that is, $\textrm{Sing}(X)$ has codimension at least $2$). (2) If $x \in X$ is terminal (resp. canonical) then it is a cyclic quotient of an index $1$ terminal (resp. canonical) singularity $0 \in Y$ by an action which is fixed point free in codimension $2$ (that is, $\textrm{Sing}(X)$ has codimension at least $3$). 

