Multi-stable morphing structures can be self-locked, easily actuated, and generate impulsive motion, and origami mechanisms with thick panels have the practical combination of morphing ability and load-bearing capacity. In this paper, we combine the two and propose a method to design multi-stable thick-origami (MSTO) structures, where the stable configurations can be arbitrarily prescribed. Bi-stable degree-4 and tri-stable degree-6 vertices with the corresponding tessellations are designed, analyzed, manufactured, and experimentally tested. Compared to the original one-degree-of-freedom thick-origami mechanisms, the multi-stable/multi-compatible version in this paper has a larger design space with more achievable configurations due to the relaxation of compatibility constraints. In addition to the existent multi-stable origami designs, such as Kresling-ori tubes and vertex-inverting locking designs, this work provides a new form of multi-stable origami with 2 to 3 arbitrarily prescribable stable configurations. Preliminary applications like self-locking deployable furniture and an impulsive gripper are demonstrated, which suggest a promising and potential prospect in many engineering fields such as deployable architectures and robotic devices.