We describe discrete maximal surfaces with singularities in 3-dimensional Minkowski space and give a Weierstrass type representation for them. In the smooth case, maximal surfaces (spacelike surfaces with mean curvature identically 0) in Minkowski 3-space generally have certain singularities. We give a criterion that naturally describes the "singular set" for discrete maximal surfaces, including a classification of the various types of singularities that are possible in the discrete case. (C) 2015 Elsevier B.V. All rights reserved.