When does a moduli space admit a spin structure? This is a very vague question. 
Is there any example of spin structures on a moduli space? References are requested.
I have vaguely heard that Witten discussed when a sigma model is spin. Somehow I cannot locate it in the literature.
Thanks!
 A: You didn't specify what kind of moduli space you meant, but there certainly are examples coming from gauge theory. The paper Spin structures on the Seiberg-Witten moduli spaces, by H. Sasahira shows that under some topological hypotheses on a 4-manifold M, the Seiberg-Witten moduli space has a spin structure.  There are similar results about the anti-self-dual Yang-Mills moduli space, but I would have to dig around to get references.
In general in the gauge theory setting, $w_2$ of the moduli space pulls back from the space of connections mod gauge group. The tangent space to the solutions to the equations (SW or ASD Yang-Mills, or….) is defined as the kernel of the linearization of the operator defining the the equations.  It follows that there is an element (the index bundle) of the K-theory of space of connections/gauge group, whose restriction to the moduli space is the tangent bundle. In particular,  $w_2$ of the tangent bundle pulls back from $w_2$ of this index bundle.  So in cases where $H^2$ with $Z_2$ coefficients for the space of connections/gauge vanishes, it follows that the moduli space is spin.  I think this happens, eg for $M=S^4$.
