Example of torsion-free sheaf which is not locally free

Question

What is a standard example of a torsion-free sheaf, say on the complex projective plane, which is not locally free?

Answer 1

A good example is the ideal sheaf of a point on $\Bbb{P}^2$, or equivalently the ideal sheaf $\mathcal{I}$ of $(0,0)\in \Bbb{A}^2$. This is torsion free because $\mathcal{I}\subset \mathcal{O}$. It is not locally free because of the answers here.

Answer 2

Another example is the function field, since projective space is integral, it has the well defined function field say $k(X)$, it defines a constant sheaf $\mathcal{K}$ such that $$\mathcal{K}(U) = k(X)$$

It is torsion free sheaf since for $0\ne a \in k(X)$, if $g \in \mathcal{O}(V)$ we have $ga = 0$ then $ga^{-1}a = g =0$ .

And the constant sheaf is not locally free.