Let $V$ be a smooth manifold, $E \rightarrow V$ a vector bundle over $V$ and $\Gamma$ be a finite group acting nontrivially on $V$ and $E$. Let $s \in C^\infty(E)$ be a generic $\Gamma$-equivariant section.

Genericity of $s$ implies that the zero locus $Z := s^{-1}(0)$ of $s$ is a singular manifold.

The order of vanishing of $s$ determines a stratification of $Z$ into bits $Z_k$, where $Z_k$ is given by all $z \in Z$, such that $s$ vanishes of order $k$ at $z$, that is $$Z_k = \lbrace z \in Z: \nabla s(z) =0, \dots, \nabla^{k}s(z) =0, \nabla^{k+1}(z) \neq 0 \rbrace$$

My Question is: are the $Z_k$'s nonisngular manifolds? To be more precise, I am interested in the $Z_k$-bit with $k$ largest so that $Z_k \neq \emptyset$.

My feeling is that $Z_k$ does not necesseraly have to be a (nonsingular) manifold, because it could consist of many complicated different bits, but I couldn't come up with a proof.

Any References would also be appreciated.