The Renyi’s information dimension (RID) of an n-dimensional random vector is the average dimension of the vector when accounting for non-zero probability measures over lower-dimensional subsets. In an information-theoretical perspective, the RID is a general measure of compressibility, beyond the continuous or discrete case. While the RID for continuous and discrete measures is well-understood, the case of a discrete-continuous measure presents a number of interesting subtleties. In this paper, we investigate the RID for a class of multi-dimensional discrete-continuous random measures with singularities on affine lower-dimensional subsets. This class of random vectors (RVs), which we term affinely singular RVs, arises from the linear transformation of orthogonally singular RVs, that RVs with singularity on affine subsets parallel to principal axes. We obtain the RID of affinely singular RVs and derive an upper bound for the RID of Lipschitz functions of orthogonally singular RVs. As an application of our results, we consider the example of a moving-average stochastic process with discrete-continuous excitation noise and obtain the block-average RID for samples of this process. We also provide insights into the relationship between the block-average dimension of the truncated samples, the minimum achievable compression rate, and other measures of compressibility for this process.