  Coherent-state superpositions are of great importance for many quantum
subjects, ranging from foundational to technological, e.g., from tests of
collapse models to quantum metrology. Here we explore various aspects of these
states, related to the connection between sub-Planck structures present in
their Wigner function and their sensitivity to displacements (ultimately
determining their metrological potential). We review this for the usual
Heisenberg-Weyl algebra associated to a harmonic oscillator, and extend it and
find analogous results for the $\mathfrak{su}(2)$ algebra, typically associated
with angular momentum. In particular, in the Heisenberg-Weyl case, we identify
phase-space structures with support smaller than the Planck action both in
cat-state mixtures and superpositions, the latter known as compass states.
However, as compared to coherent states, compass states are shown to have
$\sqrt{N}$-enhanced sensitivity against displacements in all phase-space
directions ($N$ is the average number of quanta), whereas cat states and cat
mixtures show such enhanced sensitivity only for displacements along specific
directions. We then show that these same properties apply for analogous SU(2)
states provided that (i) coherent states are restricted to the equator of the
sphere that plays the role of phase space for this group, (ii) we associate the
role of the Planck action to the `size' of SU(2) coherent states in such
sphere, and (iii) the role of $N$ to the total angular momentum.
