  We prove that homological stability fails for the moduli space of any
simply-connected closed smooth 4-manifold in any degree of homology, unlike
what happens in all dimensions $\neq 4$. We detect also the homological
discrepancy between various moduli spaces, such as topological and smooth
moduli spaces of 4-manifolds, and moduli spaces of 4-manifolds with positive
scalar curvature metrics. To prove these results, we use the Seiberg-Witten
equations to construct a new characteristic class of families of 4-manifolds,
which is unstable and detects the difference between the smooth and topological
categories in dimension 4.
