Let $C$ be a cartier divisor on a smooth projective surface in $\mathbb{P}^3$. Then we get the short exact sequence $$0 \to \mathcal{O}_X(C) \to \mathcal{O}_X(C_{red}) \to F \to 0$$ for some sheaf $F$. We see that $F$ is supported on $C$. Assuming $C \not= C_{red}$ when is it possible to say that $Ext^2_X(F,\mathcal{O}_X)=0$?

There is a long exact sequence $$ H^1(X,O_X(C_{red})) \to H^1(X,O_X(C)) \to Ext^2(F,O_X)\to H^2(X,O_X(C_{red})) \to H^2(X,O_X(C)), $$ so $Ext^2(F,O_X) = 0$ if and only if the first map is surjective and the last map is injective. 


Maybe something which can help : use Serre duality on X in order to obtain a $H^{0}$. As F is supported on C, one can compute the $H^{0}$ on C and use the adjunction formula to express the dualizing sheaf of X in terms of the dualizing sheaf of C. The conclusion is that the Ext group is zero if and only if $H^{0}(C, F(C) \otimes \omega_{C}^{1})$ is zero. 

