Set $M_k:=V(I_k) \subset \operatorname{Mat}(m \times n) \simeq \mathbb{A}^{mn}$.
(1) If $k < \min\{m, \, n\}$ then $M_{k-1}$ is precisely the singular locus of $M_k$. See
E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris: Geometry of Algebraic Curves, p. 69.
(2) The determinantal variety $M_k$ is the $k$th secant variety to $M_1$, see MSE question 4384405.