  We perform extensive classifications of $\mathbb{Z}_2$ quantum spin liquids
on the simple cubic, body centered cubic, and face centered cubic lattices
using a spin-rotation invariant fermionic projective symmetry group approach.
Taking into account that all three lattices share the same point group $O_h$,
we apply an efficient gauge where the classification for the simple cubic
lattice can be partially carried over to the other two lattices. We identify
hundreds of projective representations for each of the three lattices, however,
when constructing short-range mean-field models for the fermionic partons
(spinons) these phases collapse to only very few relevant cases. We
self-consistently calculate the corresponding mean-field parameters for
frustrated Heisenberg models on all three lattices with up to third neighbor
spin interactions and discuss the spinon dispersions, ground state energies and
dynamical spin structure factors. Our results indicate that phases with
non-uniform spinon hopping or pairing amplitudes are energetically favored. An
unusual situation is identified for the fcc lattice where the spinon dispersion
minimizing the mean-field energy features a network of symmetry protected
line-like zero modes in reciprocal space. We further discuss characteristic
fingerprints of these phases in the dynamical spin structure factor which may
help to identify and distinguish them in future numerical or experimental
studies.
