How to spot a moduli space? This is admittedly a vague question. I wonder if there is a way - a criterion - to tell if an algebraic scheme/stack $M$ is a (fine) moduli space, i.e., if parametrizes all families of isomorphism classes of algebro-geometric objects of a certain kind. 
By definition it must allow a map from the "universal family" of such objects, and entertain maps from families of the same over arbitrary bases such that pullback of the universal family agrees with the family over the base. This is a strong constraint; however, if all one has to begin with is $M$, can there be any way at all to tell that this collection of compatible maps exists? 
Even if we give ourselves a morphism $U \rightarrow M$ at the outset, can there be a criterion whereby $M$ turns out to be a moduli space with universal family $U$? In this case, we have the fibres to work with, so it's conceivable.
 A: Given that there are schemes with geometrically good properties, like the Grassmannian, that are fine moduli spaces, perhaps you want some criterion that reads like "if the scheme is this bad, then it cannot be a fine moduli space".  
Grothendieck (1966) proved that the Hilbert scheme of subschemes (with fixed Hilbert polynomial) of $P^r$ is a fine moduli space.
However, in Moduli of Curves by Harris and Morrison, we read (Murphy's Law for Hilbert Schemes):
"There is no geometric possibility so horrible that it cannot be found generically on some component of some Hilbert scheme."
So if we take this quote seriously, it is hard to imagine some general criterion that would tell you that a scheme cannot be a fine moduli space.
A: You cannot, even for some very natural moduli spaces. Take a look for instance here:
M. Kapovich, J. Millson, On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties, Math. Publications of IHES, Vol. 88 (1999) p. 5-95.
M. Kapovich, J. Millson, Universality theorem for configuration spaces of planar linkages, Topology, Vol. 41 (2002), no. 6,  p. 1051--1107.
R. Vakil, Murphy's Law in algebraic geometry: Badly-behaved deformation spaces, Invent. Math. Vol. 164 (2006), 569-590.
M. Reineke, Every projective variety is a quiver Grassmannian, Algebras and Representation Theory, Vol. 16 (2013) 1313–1314. 
