While the ground state configuration is classically determined by the variational principle minimizing the binding energy of the system, we propose here a different procedure to identify the configuration of the ground state in odd-$A$ nuclei. This procedure is based on the Hartree-Fock-Bogolyubov (HFB) framework with a self-consistent blocking of the unpaired nucleon and identifies the ground state as the blocked quasiparticle configuration compatible with the observed spin and parity and, most importantly, the measured magnetic moment. The magnetic moments are calculated within the HFB framework for all odd Hg isotopes for which experimental data is available. To validate the method, a systematic comparison between the predicted and measured electric quadrupole moments and isotopic shifts is performed. For even-even isotopes, we show that the ground state deformation, and more particularly the competition between the prolate and oblate shapes, can hardly be determined through a comparison of the QRPA $\beta$-decay half-life with experiment, though this approach also calls for an observable similar to the one used for odd-$A$ isotopes, namely the spin-flip component of the isovector part of the magnetic moment operator. For even isotopes with shape coexistence, no adequate constraint could be identified, except though the charge radius. Assuming light even-even Hg isotopes to have an oblate shape, the resulting charge radii staggering observed in the Hg chain by laser spectroscopy can through this identification procedure be reproduced.