Kaneyama and Klyachko have shown that any torus equivariant vector bundle of rank r over \(\mathbb {C}\mathbb {P}^n\) splits if \(r < n\) . In particular, any such bundle is not slope stable. In contrast, we provide explicit examples of stable equivariant reflexive sheaves of rank r on any polarised toric variety (X, L), for \(2\le r< \textrm{dim}(X)+\textrm{rank}(\textrm{Pic}(X))\) , and show that the dimension of their singular locus is strictly bounded by \(n-r\) .
