  We show that several important normal subgroups $\Gamma$ of the mapping class
group of a surface satisfy the following property: any free, ergodic,
probability measure preserving action $\Gamma \curvearrowright X$ is stably
OE-superrigid. These include the central quotients of most surface braid groups
and most Torelli groups and Johnson kernels. In addition, we show that all
these groups satisfy the measure equivalence rigidity and we describe all their
lattice-embeddings.
  Using these results in combination with previous results from [CIK13] we
deduce that any free, ergodic, probability measure preserving action of almost
any surface braid group is stably W*-superrigid, i.e., it can be completely
reconstructed from its von Neumann algebra.
