  Combining results of Wahl, Galatius--Madsen--Tillmann--Weiss and Korkmaz one
can identify the homotopy-type of the classifying space of the stable
non-orientable mapping class group $N_\infty$ (after plus-construction). At odd
primes p, the F_p-homology coincides with that of $Q_0(HP^\infty_+)$, but at
the prime 2 the result is less clear. We identify the F_2-homology as a Hopf
algebra in terms of the homology of well-known spaces. As an application we
tabulate the integral stable homology of $N_\infty$ in degrees up to six.
  As in the oriented case, not all of these cohomology classes have a geometric
interpretation. We determine a polynomial subalgebra of $H^*(N_\infty ; F_2)$
consisting of geometrically-defined characteristic classes.
