On a compact Riemannian manifold, we present stability results and their instability counterpart for sign-changing solutions of elliptic equations with critical nonlinearities. Such equations are said to be stable if any family of solutions, bounded in the energy space, is in fact bounded in C^0. In particular, we emphasize the threshold role played by the geometric Yamabe potential. As an application of stability, we obtain multiplicity results of sign-changing solutions.