If you pack hundreds of equal disks in $S^2$, typically you get hexagonal close-packing with twelve defects (points with five-fold local symmetry, rather than six-fold) corresponding to the vertices of an icosahedron.
I would guess that a packing of thousands of equal balls in $S^3$ typically has regions of face-centred cubic lattice; but what are the defects like? Are they concentrated at the vertices of a {3,3,5} or {3,4,3} polytope, or along its edges, or what?