A notion of forcing $P$ is called stationary set preserving iff each stationary subset of $\omega_1$ remains stationary in $V^P$. It is standard to show that semiproper (and of course proper) notions of forcing are stationary set preserving. On the other hand Shelah realized that assuming the Semiproper Forcing Axiom (which is consistent relative to a supercompact cardinal) each stationary set preserving notion is already semiproper. So the question naturally arises if there is a nice example of a notion of forcing which is stationary set preserving but not semiproper?
Philipp Doebler and me showed in https://ivv5hpp.uni-muenster.de/u/rds/doebler_sch.pdf that the forcing to increase u2 (which is stationary set preserving) is semi-proper iff all stationary set preserving forcings are semi-proper. So that forcing provides a canonical example.