The problem of the existence of a unique solution of a discrete-time Markov jump linear singular systems (MJLSS) and the Almost sure stability (ASS) of these systems are addressed in this paper. Based on the concept of regularity mode-to-mode, a sufficient condition for the existence and uniqueness of a solution of a certain class of MJLSS is given. Concerning the stability, a new concept of top Lyapunov exponent for an MJLSS is proposed and it is used to obtain a test to check the stability of the system.