In this paper, the regularity and stability analysis of discrete-time Markov jump linear singular systems (MJLSS) is performed. When dealing with singular systems, a primary concern is related to the existence and uniqueness of a solution to the system. This problem, that is called regularity problem, has a known solution when the linear singular system is not subject to jumps (LSS). It turns out that when the pair of matrices that describes the dynamics of the LSS satisfies a certain condition, then it is regular. By extending this condition to MJLSS, a unique solution for this class of systems is derived. Indeed through the idea of mode-to-mode regularity, which is stronger than the mode-by-mode notion that can be found in the literature, the existence of a unique solution of an MJLSS is shown. Furthermore, for systems that are mode-to-mode regular, it is shown how to obtain recursive second moment models whose complexity depends on how anticipative the systems is. The derived results on regularity and second moment modeling enable us to study stability of an MJLSS. By following the literature on Markov jump linear systems (MJLS), four different concepts of stability are introduced and their relationship is established. The results presented in this paper generalize well known results concerning the stability given in the MJLS theory. For instance, it is shown that the stability of the system is equivalent to the spectral radius of an augmented matrix being less than one, as happens in the theory of MJLS.