Reflexive sheaf on normal surfaces

Question

Let $X$ be a normal, projective scheme of pure dimension $2$ and $\mathcal{F}$ is a reflexive coherent sheaf on $X$. Is $\mathcal{F}$ locally free?

Answer 1

No. In fact, rank 1 reflexive $\mathcal{O}_X$-modules correspond to linear equivalence classes of Weil divisors, while the locally free ones correspond to Cartier divisors -- see e.g. the Appendix to §1 in Reid's Canonical 3-folds. If for instance $X$ is a quadratic cone, the reflexive rank 1 sheaf associated to a generatrix is not locally free.