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I heard from someone that, if $x:\text{Spec}k\to\mathcal{X}$ is a point of an Deligne-Mumford stack (algebraic should be OK, I am only assuming Deligne-Mumford so we know the residual gerbe exists), then the residual gerbe $\mathcal{Z}_x$ is étale-locally the classifying stack $\mathcal{B}_k\underline{ \text{Aut}}_k(x)$, i.e. the stack $[\text{Spec}k/\underline{ \text{Aut}}_k(x)]$, with the trivial action of $\underline{ \text{Aut}}_k(x)$, which is the automorphism group scheme of the point $x$.

The same person told me that if the residual gerbe is "split" or "neutralized", then in fact it is the classifying stack.

I cannot find a reference for the residual gerbe being étale-locally a classifying stack, nor can I even find a definition of what a "split" or "neutralized" gerbe is (perhaps they mean neutral? i.e. there is a section of $\mathcal{Z}_x \to Z$, where $Z$ is the algebraic space over which $\mathcal{Z}_x$ is a gerbe?) Sadly, I cannot simply ask this person, as they are not available for the next little while. Does anyone know anything about the above?

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Let $\mathcal{X}$ be the stack in question, and for simplicity, assume $\mathcal{X}$ is defined over $k$. Let $x:\mathrm{Spec}(k)\to \mathcal{X}$ a point. A residual gerbe is a gerbe, which is the content of stacks project Lemma 06QK. So the residual gerbe, being a gerbe over a field $k$, has a section after a finite extension of $k$, say $k'/k$. A gerbe with a section is a classifying stack, say $BG$ over $k'$, and this is proved in stacks project 06QG. By stacks project Lemma 06MW, the morphism $x$ factors through the residual gerbe, i.e., $x$ can be expressed as $\mathrm{Spec}(k)\to\mathcal{G}_x\to \mathcal{X}$. Now $(\mathcal{G}_x)_{k'}\cong BG$, and since $\mathcal{G}_x\to \mathcal{X}$ is a monomorphism (stacks project Lemma 06MT), $BG\cong (\mathcal{G}_x)_{k'}\to \mathcal{X}_{k'}$ is still a monomorphism, so by stacks project Lemma 06R5, $G$ is the stabilizer group of the point $x_{k'}: \mathrm{Spec}(k)\to \mathcal{X}_{k'}$. This means that the residual gerbe is a classifying stack of the stabilizer group after a finite extension of $k$, which is the meaning of "etale local" for scheme over a field.

This also answers your second question, namely, every neutral residual gerbe is a classifying stack.

mhahthhh
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