  We study the ability of variational approaches based on self-consistent
mean-field and beyond-mean-field methods to reproduce exact energies and
electromagnetic properties of the nuclei defined within the $sd$-shell valence
space using the non-trivial USD Hamiltonian. In particular,
Hartree-Fock-Bogoliubov (HFB), variation after particle-number projection
(VAPNP) and projected generator coordinate methods (PGCM) are compared to exact
solutions {obtained by} the full diagonalization of the Hamiltonian. We analyze
the role played by the proton-neutron ($pn$) mixing as well as the quadrupole
and pairing degrees of freedom (including both isoscalar and isovector
channels) in the description of the spectra of even-even, even-odd and odd-odd
nuclei in the whole $sd$-shell.
