  Background: The mercury isotopes around $N=104$ are a well-known example of
nuclei exhibiting shape coexistence. Mixing of configurations can be studied by
measuring the monopole strength $\rho^2(E0)$, however, currently the
experimental information is scarce and lacks precision, especially for the
$I^\pi \rightarrow I^\pi$ ($I \neq 0$) transitions. Purpose: The goals of this
study were to increase the precision of the known branching ratios and internal
conversion coefficients, to increase the amount of available information
regarding excited states in $^{182,184,186}$Hg and to interpret the results in
the framework of shape coexistence using different models. Method: The
low-energy structures in $^{182,184,186}$Hg were populated in the $\beta$ decay
of $^{182,184,186}$Tl, produced at ISOLDE and purified by laser ionization and
mass separation. The $\gamma$-ray and internal conversion electron events were
detected by five germanium clover detectors and a segmented silicon detector,
respectively, and correlated in time to build decay schemes. Results: In total,
193, 178 and 156 transitions, including 144, 140 and 108 observed for the first
time in a $\beta$-decay experiment, were assigned to $^{182,184,186}$Hg,
respectively. Internal conversion coefficients were determined for 23
transitions, out of which 12 had an $E0$ component. Extracted branching ratios
allowed the sign of the interference term in $^{182}$Hg as well as
$\rho^2(E0;0^+_2\rightarrow 0^+_1)$ and $B(E2;0^+_2\rightarrow 2^+_1)$ in
$^{184}$Hg to be determined. By means of electron-electron coincidences, the
$0^+_3$ state was identified in $^{184}$Hg. The experimental results were
qualitatively reproduced by five theoretical approaches, the IBM with
configuration mixing with two different parametrizations, the General Bohr
Hamiltonian, the BMF model and the SCCM model. However, a quantitative
description is lacking.
