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Let $G$ be a complex reductive group and $V$ a $G$-module. There is a naturalmoment mapping $\mu\colon V\oplus V^*\to\mathfrak{g}^*$ and we denote$\mu^{-1}(0)$ (the shell) by $N_V$. We use invariant theory and results ofMusta\c{t}\u{a} [Mus01] to find criteria for $N_V$ to have rationalsingularities and for the categorical quotient $N_V /\!\!/ G$ to havesymplectic singularities, the latter results improving upon [HSS20]. It turnsout that for ``most'' $G$-modules $V$, the shell $N_V$ has rationalsingularities. For the case of direct sums of classical representations of theclassical groups, $N_V$ has rational singularities and $N_V /\!\!/ G$ hassymplectic singularities if $N_V$ is a reduced and irreducible completeintersection. Another important special case is $V=p\,\mathfrak{g}$ (the directsum of $p$ copies of the Lie algebra of $G$) where $p\geq 2$. We show that$N_V$ has rational singularities and that $N_V /\!\!/ G$ has symplecticsingularities, improving upon results of [Bud19], [AA16], [Kap19] and [GH20].Let $\pi=\pi_1(\Sigma)$ where $\Sigma$ is a closed Riemann surface of genus$p\geq 2$. Let $G$ be semisimple and let $\operatorname{Hom}(\pi,G)$ and$\mathscr X\!(\pi,G)$ be the corresponding representation variety and charactervariety. We show that $\operatorname{Hom}(\pi,G)$ is a complete intersectionwith rational singularities and that $\mathscr X\!(\pi,G)$ has symplecticsingularities. If $p>2$ or $G$ contains no simple factor of rank $1$, then thesingularities of $\operatorname{Hom}(\pi,G)$ and $\mathscr X\!(\pi,G)$ are incodimension at least four and $\operatorname{Hom}(\pi,G)$ is locally factorial.If, in addition, $G$ is simply connected, then $\mathscr X\!(\pi,G)$ is locallyfactorial.

When does the zero fiber of the moment map have rational singularities?