Let $X$ be an Artin stack over the complex numbers. What can one say about the local structure of $X$, i.e. what is the simplest class of stacks by which were can always find a cover of $X$ by open substacks in this class? For example, does $X$ always have a cover by open substacks of the form $U/G$ where $G$ is an algebraic group acting on a scheme or complex analytic space $U$? I actually suspect that this is false, and that a counterexample is given by the moduli stack of nodal genus zero curves (I don't see how to construct such a neighborhood around $\mathbb CP^1\vee\mathbb CP^1$). If it is indeed false, what is the best one can say? I am very willing to pass to the analytification of $X$ if this helps get a stronger local structure result.

A related question: Stacks as local quotients or via atlases