Let F be either R or a finite extension of Qp, and let G be a finite central extension of the group of F -points of a reductive group defined over F . Also let π be a smooth representation of G (Fréchet of moderate growth if F = R). For each nilpotent orbit O we consider a certain Whittaker quotient πO of π. We define the Whittaker support WS(π) to be the set of maximal O among those for which πO = 0.In this paper we prove that all O ∈ WS(π) are quasi-admissible nilpotent orbits, generalizing some of the results in [Moe96, JLS16]. If F is p-adic and π is quasi-cuspidal then we show that all O ∈ WS(π) 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 a conjecture from [Gin06].Our methods are a synergy of the methods of the above-mentioned papers, and of our preceding paper [GGS17].