  We study the wave-front set of an element in a $p$-adic reductive Lie algebra
(for $p\gg\operatorname{rank}$), namely the set of maximal nilpotent orbits
appearing in its Shalika germ expansion. By adapting an algorithm of
Waldspurger that computes orbital integrals, we obtain an inductive algorithm
to compute an invariant that determines the wave-front set. This gives an
algorithm to compute the wave-front sets for regular supercuspidal
representations and reveals examples whose wave-front sets are not contained in
a single geometric orbit, for arbitrarily large $p$ within a fixed rank.
