Question: Is the mapping class group of a surface virtually torsion-free?
Context: The mapping class group of the $ n $ punctured disk is the braid group $ B_n $. The braid group is torsion-free and thus virtually torsion-free. The mapping class group of the torus is $ SL(2,\mathbb{Z}) $ which is virtually torsion-free and indeed virtually free (has an index 12 free subgroup https://mathoverflow.net/questions/43726/the-free-group-f-2-has-index-12-in-sl2-mathbbz). Note that braid groups are not virtually free, except $ B_2 \cong \mathbb{Z} $, see Subgroups of the braid groups $\mathcal{B}_n$ .