  We prove that regular supercuspidal representations of $p$-adic groups are
uniquely determined by their character values on very regular elements -- a
special class of regular semisimple elements on which character formulae are
very simple -- provided that this locus is sufficiently large. As a
consequence, we resolve a question of Kaletha by giving a description of
Kaletha's $L$-packets of regular supercuspidal representations which mirrors
Langlands' construction for real groups following Harish-Chandra's
characterization theorem for discrete series representations. Our techniques
additionally characterize supercuspidal representations in general, giving
$p$-adic analogues of results of Lusztig on reductive groups over finite
fields. In particular, we establish an easy, non-cohomological characterization
of unipotent supercuspidal representations when the residue field of the base
field is sufficiently large.
