  Let $\epsilon_1, \dotsc, \epsilon_n$ be i.i.d. Rademacher random variables
taking values $\pm 1$ with probability $1/2$ each. Given an integer vector
$\boldsymbol{a} = (a_1, \dotsc, a_n)$, its concentration probability is the
quantity $\rho(\boldsymbol{a}):=\sup_{x\in \mathbb{Z}}\Pr(\epsilon_1
a_1+\dots+\epsilon_n a_n = x)$. The Littlewood-Offord problem asks for bounds
on $\rho(\boldsymbol{a})$ under various hypotheses on $\boldsymbol{a}$, whereas
the inverse Littlewood-Offord problem, posed by Tao and Vu, asks for a
characterization of all vectors $\boldsymbol{a}$ for which
$\rho(\boldsymbol{a})$ is large. In this paper, we study the associated
counting problem: How many integer vectors $\boldsymbol{a}$ belonging to a
specified set have large $\rho(\boldsymbol{a})$? The motivation for our study
is that in typical applications, the inverse Littlewood-Offord theorems are
only used to obtain such counting estimates. Using a more direct approach, we
obtain significantly better bounds for this problem than those obtained using
the inverse Littlewood--Offord theorems of Tao and Vu and of Nguyen and Vu.
Moreover, we develop a framework for deriving upper bounds on the probability
of singularity of random discrete matrices that utilizes our counting result.
To illustrate the methods, we present the first `exponential-type' (i.e.,
$\exp(-n^c)$ for some positive constant $c$) upper bounds on the singularity
probability for the following two models: (i) adjacency matrices of dense
signed random regular digraphs, for which the previous best known bound is
$O(n^{-1/4})$ due to Cook; and (ii) dense row-regular $\{0,1\}$-matrices, for
which the previous best known bound is $O_{C}(n^{-C})$ for any constant $C>0$
due to Nguyen.
