A plane set admits an inscribed rectangle if every homeomorphic copy of it in $\mathbb{R}^2$ contains the 4 vertices of at least one Euclidean rectangle. Vaughan proved that $S^1$ admits an inscribed rectangle by reducing the problem to the non-embeddability of the projective plane in $\mathbb{R}^3$ (it is not known if $S^1$ admits an inscribed rectangle of aspect ratio 1:1 i.e. a square). In this talk, using the non-embeddability of the Cone($K_5$) and the Cone($K_{3,3}$) in $\mathbb{R}^3$ we classify plane compact connected locally-connected sets that admit inscribed rectangles. Using similar topological techniques, we also present a one-dimensional non-connected set such that every copy of it in $\mathbb{R}^2$ admits an inscribed rectangle with at least one vertex in each component.