TITLE: examples of moduli functors for which coarse moduli space does not exists
QUESTION [6 upvotes]: 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.

REPLY [6 votes]: Here is a classifying problem without a coarse moduli space:  Let $F$ be the stack of line bundles with section, say over an algebraically closed field k.  That is $F(X)$ is the category of pairs $(L,s)$ where $L$ is a line bundle on $X$ and $s \in \Gamma(X,L)$ is a section of $L$.  Then $F$ has no coarse moduli space:  if it did, there would be two points, corresponding to $(k,1)$ and $(k,0)$ with the former specializing to the latter.  This would be an affine scheme with a closed point and a generic point, both with residue field $k$.  That's impossible.
Of course, $F$ is just a geometric description of $[\mathbf{A}^1 / \mathbf{G}_m]$, so we already knew it didn't have a coarse moduli space.