We introduce the notion of linear multifractional stable sheets in the broad sense (LMSS) to include both linear multifractional Brownian sheets and linear multifractional stable sheets. The purpose of the present paper is to study the existence and joint continuity of the local times of LMSS, and also the local Holder condition of the local times in the set variable. As the main results, (1) we provide a sufficient and necessary condition for the existence of local times of LMSS; (2) we show a sufficient condition for the joint continuity of local times; and (3) we prove a sharp local Holder condition for the local times in the set variable. All these fruits improve significantly the existing results for the local times of multifractional Brownian sheets and linear multifractional stable sheets in the literature.