  We have recently proved a homological stability theorem for moduli spaces of
r-Spin Riemann surfaces, which in particular implies a Madsen--Weiss theorem
for these moduli spaces. This allows us to effectively study their stable
cohomology, and to compute their stable rational cohomology and their integral
Picard groups. Using these methods we give a complete description of their
integral Picard groups for genus at least 9 in terms of geometrically defined
generators, and determine the relations between them.
