In this article we develop new and very general techniques for the analysis of the stability properties of finite-difference approximations of elastic wave equations, as used in solids and liquids. It is shown that if certain common conditions are satisfied, then the time dependence of every permitted grid mode in any of these formulations can be related to a single time-independent eigenvalue. It is further shown that only a few (generally two or three) modes are relevant to stability and that for these modes the values of the time-independent eigenvalue can be determined by simple numerical experiments. These results permit a simple and thorough analysis of stability to be undertaken, both as a preliminary to making measurements using a particular formulation and as a means of comparing different formulations. The analysis is validated, and its application is demonstrated using two examples, taken from a new twin-grid formulation for the solid–liquid interface and from the Composed Approximation for a solid free surface.