I am studying reflexive sheaves (on $\mathbb{P}^3$) by the Hartshorne's paper ''Stable reflexive sheaves''. As far I understood, reflexive sheaves fail to be locally free at a finite number of points (on $\mathbb{P}^3$, for higher dimensional varieties it fail to be locally free in a scheme of codimension 3). Therefore, if one restricts a reflexive sheaf in an open of $\mathbb{P}^3$ not containing the singular points, the restricted sheaf should isomorphic to copies of the trivial. I would like to understand what happens when one restrict the sheaf to an open containing the singular point. Is there a way of computing explicit what the restricted sheaf will look like?

Thank you in advance.