In a survey paper of Korkmaz it is stated that $H_2(\mathrm{Mod}_3)$ is either $\Bbb Z$ or $\Bbb Z \oplus \Bbb Z_2$, but I was not able to find out a precise computation of this group (resolving the ambiguity). Is this group known? Any reference?

$\begingroup$ Did you have a look at Chapter 12 of sciencedirect.com/science/book/9780444824325 (Ivanov's survey on mapping class groups)? I don't have a copy right now, so can not check, but usually the book contains everything one would like to know about the homology of mapping class groups. $\endgroup$– ThiKuAug 24 '14 at 6:02

2$\begingroup$ @ThiKu : while I think that Ivanov's survey is great, it is pretty dated at this point (eg it predates MadsenWeiss). $\endgroup$– Andy PutmanAug 24 '14 at 13:44

$\begingroup$ @ThiKu : I gave a look at Ivanov's survey, but it consider this particular case with rational coefficinets. As in Andy's answer, the integral second homology of $\mathrm{Mod}_3$ has been computed recently by Sakasai. Anyway, thank you for your comment! $\endgroup$– Daniele ZuddasAug 25 '14 at 15:24
In his paper "Lagrangian mapping class groups from a group homological point of view" (available here), Sakasai proves that the desired homology group is $\mathbb{Z} \oplus \mathbb{Z}/2$.

$\begingroup$ Sorry, typed this on my phone and will not have access to a computer soon. Can someone fix the arxiv link and the tex? Thanks! $\endgroup$ Aug 24 '14 at 13:46