  Let $F/\mathbb{Q}_p$ be finite and let $\mathfrak{X}_G$ be the moduli space
of Langlands parameters valued in $G$, in characteristic distinct from $p$.
First, we determine the irreducible components of $\mathfrak{X}_G$. Then, we
determine the local structure around tamely ramified points for which the image
of `tame inertia' is regular. This local structure is related to the
endomorphism rings of Gelfand--Graev representations, by work of Li. Lastly, we
determine an open dense set in $\mathfrak{X}_M$, when $M$ is a Levi subgroup of
$G$, such that the natural map of moduli stacks $[\mathfrak{X}_M/M] \to
[\mathfrak{X}_G/G]$ is smooth on this set.
