We present results of a linear stability analysis of relativistic deto
nation fronts, which have been considered as representing phase interf
aces in cosmological first-order phase transitions: After discussing g
eneral stability conditions for detonation fronts, we concentrate on t
he properties of the fronts with respect to corrugation instabilities
and discuss separately the cases of Chapman-Jouguet and strong detonat
ion waves. Contrary to what is recently claimed, we find that strong d
etonations are both evolutionary and stable with respect to corrugatio
ns of the front. Moreover, Chapman-Jouguet detonations appear to be un
conditionally linearly stable. The implications of the stability resul
ts for first-order cosmological phase transitions are presented and a
discussion of the causal structure of reaction fronts is also given.