We present new results on the entropy cone generated by stabilizer states. It is a long-standing open problem to characterize the marginal von Neumann entropies that can be realized in many-body quantum states. Recently, several authors started to investigate the set of marginal entropies associated with stabilizer states. They provide both an inner approximation to the full entropy cone, as well as a rich toy theory that is worth studying for its own sake. We identify new sets of inequalities respected by stabilizer states. Our main result is a partial answer to a question raised by Linden, Ruskai, and Winter. They asked whether stabilizer entropies respect certain inequalities arising from subspace arrangements, the so called linear rank inequalities. We show that this is indeed the case for square-free dimensions.