  Kagome metals $A$V$_3$Sb$_5$ ($A=$K, Rb, Cs) exhibit an exotic charge order
(CO), involving three order parameters, with broken translation and
time-reversal symmetries compatible with the presence of orbital currents. The
properties of this phase are still intensely debated, and it is unclear if the
origin of the CO is mainly due to electron-electron or electron-phonon
interactions. Most of the experimental studies confirm the nematicity of this
state, a feature that might be enhanced by electronic correlations. However, it
is still unclear whether the nematic CO becomes stable at a temperature equal
to ($T_{\text{nem}} = T_\text{C}$) or lower than ($T_{\text{nem}} <
T_\text{C}$) the one of the CO itself. Here, we systematically characterize
several CO configurations, some proposed for the new member of the family
ScV$_6$Sn$_6$, by combining phenomenological Ginzburg-Landau theories, valid
irrespective of the specific ordering mechanism, with mean-field analysis. We
find a few configurations for the CO that are in agreement with most of the
experimental findings to date and that are described by different
Ginzburg-Landau potentials. We propose to use resonant ultrasound spectroscopy
to experimentally characterize the order parameters of the CO, such as the
number of their components and their relative amplitude, and provide an
analysis of the corresponding elastic tensors. This might help understand which
mean-field configuration found in our study is the most representative for
describing the CO state of kagome metals, and it can provide information
regarding the nematicity onset temperature $T_\text{nem}$ with respect to
$T_\text{C}$.
