  We introduce mountain-pass type arguments in the context of orbital
instability for Klein-Gordon equations. Our aim is to illustrate on two
examples how these arguments can be useful to simplify proofs and derive new
results of orbital stability/instability. For a power-type nonlinearity, we
prove that the ground states of the associated stationary equation are
minimizers of the functional "action" on a wide variety of constraints. For a
general nonlinearity, we extend to the dimension N=2 the classical instability
result for stationary solutions of nonlinear Klein-Gordon equations proved in
1985 by Shatah in dimension N>2.
