I will report on joint work in progress with Akshay Venkatesh where, motivated by anomalous class group statistics, we propose an arith- metic analogue of the topological story of quadratic enhancements associ- ated with spin structures on closed oriented 2- and 3-manifolds: a choice of spin structure provides, respectively, a quadratic refinement of the mod 2 intersection form and of the linking pairing on the first torsion homology. I will put special emphasis on the implications for class group heuristics.