The aim of this paper is to present some results allowing positively invariant and asymptotically stable polyhedral sets to be determined for the linear singular discrete-time system ExK+1= AxK; the regular and singular cases of pencil (E, A) are considered. Necessary and sufficient algebraic conditions guaranteeing that some sets are positively invariant and asymptotically stable are given. Also discussed are some issues relating to the positive invariance of some domains and consistent initial conditions. Finally, as an application, the proposed results are used to determine a stability domain for a state feedback regulator with constraints on controls