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.