  We prove that the singular locus of a rank 2 instanton sheaf $E$ on
$\mathbb{P}^3$ which is not locally free has pure dimension 1. Moreover, we
also show that the dual and double dual of $E$ are isomorphic locally free
instanton sheaves, and that the sheaves
$\mathcal{E}xt^1(E,\mathcal{O}_{\mathbb{P}^3)$ and $E^{\vee\vee}/E$ are rank
$0$ instantons. We also provide explicit examples of instanton sheaves of rank
$3$ and $4$ illustrating that all of these claims are false for higher rank
instanton sheaves.
