In this article we carry out a detailed investigation of the geometric nature of the points at infinity of Minkowski superspace. It turns out that there are several sets of points forming the superconformal boundary of Minkowski superspace: on top of a well-behaved super $\mathscr{I}$, we find other sets that we exhibit and study. We also study the intersection of these boundaries with super null cones and explicitly construct the corresponding space of super cuts.