  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 to
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 in both
Schr\"{o}dinger-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
in specific directions. We then show that these same properties apply for
analogous SU(2) states provided (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 a sphere, and (iii) we associate the role of $N$ with the total angular
momentum.
