This paper considers the stability and stabilizability problems for the class of linear switching singular (LSS) systems. The stabilizing class of controllers we will be considering in this work is a constant-gain one. First, we begin by giving sufficient LMI-based conditions for stability and stabilizability of regular switching systems. Subsequently, we extend the theory to linear singular switching systems (LSS) that can be regularized by output derivative injection. Numerical examples are given to show the effectiveness of the control strategy.