  Let $F$ be either $\mathbb{R}$ or a finite extension of $\mathbb{Q}_p$, and
let $G$ be a finite central extension of the group of $F$-points of a reductive
group defined over $F$. Also let $\pi$ be a smooth representation of $G$
(Frechet of moderate growth if $F=\mathbb{R}$). For each nilpotent orbit
$\mathcal{O}$ we consider a certain Whittaker quotient $\pi_{\mathcal{O}}$ of
$\pi$. We define the Whittaker support WS$(\pi)$ to be the set of maximal
$\mathcal{O}$ among those for which $\pi_{\mathcal{O}}\neq 0$.
  In this paper we prove that all $\mathcal{O}\in\mathrm{WS}(\pi)$ are
quasi-admissible nilpotent orbits, generalizing some of the results in
[Moe96,JLS16]. If $F$ is $p$-adic and $\pi$ is quasi-cuspidal then we show that
all $\mathcal{O}\in\mathrm{WS}(\pi)$ are $F$-distinguished, i.e. do not
intersect the Lie algebra of any proper Levi subgroup of $G$ defined over $F$.
  We also give an adaptation of our argument to automorphic representations,
generalizing some results from [GRS03,Shen16,JLS16,Cai] and confirming some
conjectures from [Ginz06].
  Our methods are a synergy of the methods of the above-mentioned papers, and
of our preceding paper [GGS17].
