Dave Benson and Fred Cohen
Mapping class groups of low genus and their cohomology.
Memoirs of the AMS 443 (1991).
MR 91g:57002

\begin{abstract}
  This series of papers is aimed towards the calculation of the cohomology of 
  the mapping class group of a closed oriented surface of genus two. This is 
  all $2$, $3$ and $5$-torsion (Theorem~1.1, p.~6). The mod $5$ cohomology is 
  given in the first paper (Theorem~1.2, p.~7), the mod $3$ cohomology in the 
  second (Theorem~1.1, p.~29), and the mod $2$ cohomology in the third 
  (Theorem~1.1, p.~86). Along the way, we investigate many interesting 
  properties of mapping class groups. This involves us in the study of the 
  braid groups, modular representations of symmetric groups, and configuration 
  spaces.
\end{abstract}