ClpX and related AAA+ ATPases of the Clp/Hsp100 family are able to denature native proteins. Here, we explore the role of protein stability in ClpX denaturation and subsequent ClpP degradation of model substrates bearing ssrA degradation tags at different positions. ClpXP degraded T. thermophilus RNase-H* with a C-terminal ssrA tag very efficiently, despite the very high global stability of this thermophilic protein. In fact, global thermodynamic stability appears to play little role in susceptibility to degradation, as a far less stable RNase-H*-ssrA mutant was degraded more slowly than wild type by ClpXP and a completely unfolded mutant variant was degraded less than twice as fast as the wild-type parent. When ssrA peptide tags were covalently linked to surface cysteines at positions 114 or 140 of RNase-H*, the conjugates were proteolyzed very slowly. This resistance to degradation was not caused by inaccessibility of the ssrA tag or an inability of ClpXP to degrade proteins with side-chain linked ssrA tags. Our results support a model in which ClpX denatures proteins by initially unfolding structural elements attached to the degradation tag, suggest an important role for the position of the degradation tag and direction of force application, and correlate well with the mapping of local protein stability within RNase-H* by native-state hydrogen exchange.