Let $X$ be a smooth projective surface over $\mathbb{C}$. Suppose we have an injective morphism $$0\rightarrow L\rightarrow V$$ from a line bundle to a rank two vector bundle on $X$ such that the quotient $L/V$ is torsion-free. So $V/L$ fails to be locally free only at a finite set of points.
Suppose $C$ is smooth curve on $X$. The restriction $V/L|_C$ has generically rank one. So we get the exact sequence $$0\rightarrow L|_C\rightarrow V|_C\rightarrow V/L|_C\rightarrow 0\,.$$ Do we know that $V/L|_C$ is torsion-free as well.
That depends on whether the curve passes through the points or not. Indeed, let $F$ be the Artin sheaf $(V/L)^{\vee\vee}/(V/L)$. Tensoring it by $$ 0 \to O(-C) \to O \to O_C \to 0 $$ we obtain $$ 0 \to Tor_1(F,O_C) \to F(-C) \to F \to F \otimes O_C \to 0, $$ hence $Tor_1(F,O_C)$ is a torsion sheaf supported on the intersection of $C$ with the support of $F$. On the other hand, tensoring $$ 0 \to V/L \to (V/L)^{\vee\vee} \to F \to 0 $$ with $O_C$ we obtain $$ 0 \to Tor_1(F,O_C) \to (V/L)\vert_C \to (V/L)^{\vee\vee}\vert_C \to F \otimes O_C \to 0, $$ hence this sheaf gives the torsion part of $(V/L)\vert_C$.