We give definitions of the framed mapping class group and the Pin mapping
class groups of a smooth surface. Earlier work of the author is shown to imply
that these groups all satisfy homological stability, and we show that the
stable homology coincides with the homology of the infinite loop spaces
\Omega^\infty_0 S^{-2} and \Omega^\infty_0 MTPin(2) respectively. In
particular: the stable framed mapping class group has trivial rational
homology, and its abelianisation is Z/24; the rational homology of the stable
Pin mapping class groups coincides with that of the non-orientable mapping
class group, and their abelianisations are Z/2 for Pin^+ and (Z/2)^3 for Pin^-.