Instabilities have been commonly prevented in traditional engineering design. However, recently, origami-based instability mechanisms have been explored to design novel material or structural systems, e.g., Miura-ori-inspired morphing metamaterials, square-twist-based programmable metasheets, and Kresling origami-inspired high-damping devices. To overcome limitations of the origami design space, we seek to use the ground structure approach to explore potential crease layouts for an origami-inspired design with instabilities. The goal is to inverse-design optimal crease skeletons (truss structures) with programmable snapping instabilities. From an algorithmic viewpoint, we use a min–max topology optimization formulation subject to prescribed nonlinear structure responses (i.e., snapping equilibrium path). The objective consists of minimizing the errors between actual and prescribed load factors under given deformation. To capture the snapping equilibrium path in structural analysis, we adopt a modified generalized displacement control. We verify that the method can capture the equilibrium paths with prescribed snapping instabilities (snap-through and snap-back behaviors). Several two- and three-dimensional examples demonstrate the capabilities of the proposed formulation for programming structural instabilities. The present rod-based optimized designs can be reconfigured, which leads to desired functionalities, such as programmable snapping sequences and tunable mechanical responses.