You actually care about the number of singular matrices (which is the difference between the number of invertible matrices and the number of unrestricted matrices). This has been studied: see
Author Yonatan R. Katznelson Title: Integral Matrices of Fixed Rank Journal: proceedings of the AMS, 120(3) 1994
ADDITION It would be useful to adjoin my comments to @Gerry's answer:
The OP is NOT asking for enumeration of matrices in $SL(n, \mathbb{Z}),$ but rather for the cardinality of the intersection of $M^n(\mathbb{Z}) \cap GL(n, \mathbb{C}).$ On the other hand, the first asymptotic result for $SL(2, Z)$ I am aware of (using theta functions, with no error term) is given by Morris Newman:
Newman, Morris(1-UCSB) Counting modular matrices with specified Euclidean norm. J. Combin. Theory Ser. A 47 (1988), no. 1, 145–149.
I am unaware of the Selberg reference. However, the Newman result was generalized by Duke, Rudnick, Sarnak in Duke, W.(1-RTG); Rudnick, Z.(1-STF); Sarnak, P.(1-STF) Density of integer points on affine homogeneous varieties. Duke Math. J. 71 (1993), no. 1, 143–179. (the authors were unaware of Newman's work), with full asymptotics, and in a companion paper, a "softer" result was derived by Eskin and McMullen by ergodic-theoretic methods in the very well-known paper
Eskin, Alex(1-PRIN); McMullen, Curt(1-CA) Mixing, counting, and equidistribution in Lie groups. Duke Math. J. 71 (1993), no. 1, 181–209.
The paper of Yonatan Katznelson cited above is a sort of an off-shoot of Duke/Rudnick/Sarnak (Katznelson was a student of Sarnak, and I believe the paper was a part of his thesis).