I am trying to understand when the module $Hom(F,k)$ is non empty, where $F$ is a rank $r \geq 2$ torsion-free sheaf on a 3-fold $X$, and $k$ the skyscraper sheaf of one point on $X$.
While $Hom(F,k)$ is non empty for $F$ locally free or reflexive, this seems not to be the case for any torsion-free sheaf. Indeed, since $F$ is torsion-free and $X$ a 3-fold, there exists a locally free resolution of $F$ of the form:
$$0\to L_2 \to L_1 \to L_0 \to F$$
which applying functor $\mathcal{Hom}(,k)$ on the above sequence gives :
$\dim Hom(F,k) = r + \dim H^0(\mathcal{ext}^1(F,k)) - H^0(\mathcal{ext}^2(F,k)) $
while $H^0(\mathcal{ext}^2(F,k)) = 0$ if $F$ is reflexive or locally free, this can not be true if $F$ is only torsion-free, in this case, I don't know any condition that prevents that $\dim H^0(\mathcal{ext}^1(F,k)) - H^0(\mathcal{ext}^2(F,k)) = -r$.
But this will be very strange, so I would like to know if there is one example of torsion-free sheaf such that $Hom(F,k) = 0$.
The question is local, so you may replace $F$ by a module $M$ over a local ring $A$, and ask about $Hom(M,k)$. Note that if $\mathfrak{m} \subset A$ is the maximal ideal, then $M \ne \mathfrak{m}M$ by Nakayama Lemma, hence $M/\mathfrak{m}M$ is a non-zero vector space, so you can choose a non-zero linear function on it and define a morphism $$ M \to M/\mathfrak{m}M \to k $$ which is surjective, hence non-zero.