We consider the traveling standing waves of the Klein-Gordon equation
$u_{tt}-u_{xx}+u-|u|^{p-1} u=0$
in the whole line case as well as the periodic case. In both situations we study the stability of the waves via recently developed abstract stability criteria. The exact ranges of the wave speeds and the frequencies needed for stability are derived for all solitary waves and for some integer values of $p$ in the periodic case.