  Neutron-deficient $^{177-185}$Hg isotopes were studied using in-source laser
resonance-ionization spectroscopy at the CERN-ISOLDE radioactive ion-beam
facility, in an experiment combining different detection methods tailored to
the studied isotopes. These include either alpha-decay tagging or
Multi-reflection Time-of-Flight gating to identify the isotopes of interest.
The endpoint of the odd-even nuclear shape staggering in mercury was observed
directly by measuring for the first time the isotope shifts and hyperfine
structures of $^{177-180}$Hg. Changes in the mean-square charge radii for all
mentioned isotopes, magnetic dipole and electric quadrupole moments of the
odd-A isotopes and arguments in favor of $I = 7/2$ spin assignment for
$^{177,179}$Hg were deduced. Experimental results are compared with Density
Functional Theory (DFT) and Monte-Carlo Shell Model (MCSM) calculations. DFT
calculations with several Skyrme parameterizations predict a large jump in the
charge radius around the neutron $N = 104$ mid shell, with an odd-even
staggering pattern related to the coexistence of nearly-degenerate oblate and
prolate minima. This near-degeneracy is highly sensitive to many aspects of the
effective interaction, a fact that renders perfect agreement with experiment
out of reach for current functionals. Despite this inherent diffculty, the
SLy5s1 and a modified UNEDF1^{SO} parameterization predict a qualitatively
correct staggering that is off by two neutron numbers. MCSM calculations of
states with the experimental spins and parities show good agreement for both
electromagnetic moments and the observed charge radii. A clear mechanism for
the origin of shape staggering within this context is identified: a substantial
change in occupancy of the proton $\pi h_{9/2}$ and neutron $\nu i_{13/2}$
orbitals.
