  Various properties of single-particle Hartree-Fock ground-state solutions in
$N\sim Z$ nuclei are investigated. The emphasis is on a role of single-particle
mean-field in odd-even mass staggering. It is shown that, unlike in traditional
scenario originating from the Fermi gas or macroscopic models, the symmetry
energy contribution to odd-even mass staggering is nearly cancelled by the
contribution coming from the average level density. It allows to construct
indicators probing both pairing as well as mean-field components to the
odd-even mass staggering. The impact of the single-particle Hartree-Fock field
on Wigner energy and residual $pn$ interaction in odd-odd nuclei is also
discussed.
