In the modern formulation of lattice gauge fixing, the gauge-fixing condition is written in terms of the minima or stationary points (collectively called solutions) of a gauge-fixing functional. Due to the non-linearity of this functional, it usually has many solutions, called Gribov copies. The dependence of the number of Gribov copies, n[U], on the different gauge orbits plays an important role in constructing the Faddeev-Popov procedure and hence in realising the BRST symmetry on the lattice. Here, we initiate a study of counting n[U] for different orbits using three complimentary methods: (1) analytical results in lower dimensions, and some lower bounds on n[U] in higher dimensions, (2) the numerical Polynomial homotopy continuation method, which numerically finds all Gribov copies for a given orbit for small lattices, and (3) numerical minimisation ("brute force"), which finds many distinct Gribov copies, but not necessarily all. Because n for the coset SU(N-c)/U(1) of an SU(N-c) theory is orbit independent, we concentrate on the residual compact U(1) case in this article, and establish that n is orbit dependent for the minimal lattice Landau gauge and orbit independent for the absolute lattice Landau gauge. We also observe that, contrary to a previous claim, n is not exponentially suppressed for the recently proposed stereographic lattice Landau gauge compared to the naive gauge in more than one dimension. (C) 2013 Elsevier Inc. All rights reserved.