Some problems call for a count of the number of integer "lattice" points contained in a feasible region (rather than for locating the minimum or maximum objective function value in that region). See for example the SO Question "Counting solutions to linear inequalities" or this previous Math.SE Question "Uniform lattice sample inside a particular convex polytope".
As a general proposition this counting is difficult (NP-hard as the corresponding decision problem is NP-complete), but it is sometimes possible to count by writing custom code, as I did for this previous SciComp.SE Question "Python solvers for mixed-integer nonlinear constrained optimization".
What solvers or software packages can be used for the purpose of counting (or estimating the count of) feasible integer solutions? My understanding is that the MILP solver CPLEX has an option to collect a "solution pool" of the integer portion of feasible points it encounters in a run.
I'd also appreciate a basic example of using this feature to count or estimate the number of feasible points of an ILP. Links to such examples would be more helpful if accompanied by a discussion of the limitations or difficulties of that task. As a comment points out, CPLEX will only report the feasible points it locates during a search for optimizing an objective function, but the number of feasible points is independent of the objective function. Therefore the sampling in such solution pools is biased and incomplete. Nevertheless it might be possible to collect a representative sample by varying the objective function.
It might also be that better tools are available for the purpose, as another comment suggests.