Well, the title almost says it all. I would like to list as many examples as possible of moduli functors, for which a coarse moduli space does not exist (and maybe explain why). So, examples such as $[\mathbb{A}^1/G\_m]$ are not what I mean. I would really like to see moduli functors that come from some classifying problem.



examples of moduli functors for which coarse moduli space does not exists
$\overline{\mathfrak{M}}\_{0,0}$, the stack of pre-stable genus zero curves. This parameterizes genus zero curves having at worst nodal singularities. There are an infinite number of isomorphism classes of such curves, but they all occur as a specialization of a trivial family (just do repeated blowups of the central fiber of the family $\mathbb{P}^1\times \mathbb{A}^1 \to \mathbb{A}^1$ ). 


