We develop a general theory to treat the linear stability of certain special solutions of second order in time evolutionary PDE. We apply these results to standing waves of the following problems: the Klein-Gordon equation, for which we consider both ground states and certain excited states, the Klein-Gordon-Zakharov system and the beam equation. We also discuss applications to excited states for the Klein-Gordon model as well as multidimensional traveling waves (not necessarily homoclinic to zero) for general second order equations of this type. In all cases, our abstract results provide a complete characterization of the linear stability of such solutions. © 2013 Elsevier B.V. All rights reserved.