  The Majorana stellar representation is used to characterize spin states that
have a maximally negative Wigner quasiprobability distribution on a spherical
phase space. These maximally Wigner-negative spin states generally exhibit a
partial but not high degree of symmetry within their star configurations. In
particular, for spin $j > 2$, maximal constellations do not correspond to a
Platonic solid when available and do not follow an obvious geometric pattern as
dimension increases. In addition, they are generally different from spin states
that maximize other measures of nonclassicality such as anticoherence or
geometric entanglement. Random states ($j \leq 6$) display on average a
relatively high amount of negativity, but the extremal states and those with
similar negativity are statistically rare in Hilbert space. We also prove that
all spin coherent states of arbitrary dimension have non-zero Wigner
negativity. This offers evidence that all pure spin states also have non-zero
Wigner negativity. The results can be applied to qubit ensembles exhibiting
permutation invariance.
