  In this paper we study quantitative aspects of trace characters $\Theta_\pi$
of reductive $p$-adic groups when the representation $\pi$ varies. Our approach
is based on the local constancy of characters and we survey some other related
results. We formulate a conjecture on the behavior of $\Theta_\pi$ relative to
the formal degree of $\pi$, which we are able to prove in the case where $\pi$
is a tame supercuspidal. The proof builds on J.-K.~Yu's construction and the
structure of Moy-Prasad subgroups.
