In this paper.. the robust D-stability problem (i.e. the robust eigenvalue-clustering in a specified circular region problem) of linear discrete singular systems with both structured (elemental) parameter uncertainties and delayed perturbations is investigated. Under the assumptions that the linear discrete nominal singular system is regular and causal, and has all its finite eigenvahies lying inside a specified circular region, by using the maximum modulus principle and the spectral radius of matrices, a new sufficient condition is proposed to preserve the assumed properties when both structured parameter uncertainties and delayed perturbations are added into the linear discrete nominal singular system. When all the finite eigenvalues lie inside the unit circle of the z-plane, the proposed criterion will become the stability robustness criterion. The proposed criterion is mathematically proved and numerically shown, respectively, to be less conservative than the existing ones reported recently.