Photoneutron cross sections, including $\ensuremath{\sigma}[(\ensuremath{\gamma},n)+(\ensuremath{\gamma},pn)]$, $\ensuremath{\sigma}[(\ensuremath{\gamma},2n)+(\ensuremath{\gamma},p2n)]$, and $\ensuremath{\sigma}(\ensuremath{\gamma},3n)$, were measured for $^{188}\mathrm{Os}$, $^{189}\mathrm{Os}$, $^{190}\mathrm{Os}$, and $^{192}\mathrm{Os}$ from 7 to 30 MeV and for $^{186}\mathrm{Os}$ from 11 to 20 MeV, with a photon energy resolution of about 300 keV. The source of radiation was the monoenergetic photon beam obtained from the annihilation in flight of fast positrons. The partial photoneutron cross sections were determined by neutron multiplicity counting, and the average neutron energies for each multiplicity were determined simultaneously with the cross-section data by the ring-ratio technique. Nuclear information extracted from the data includes parameters of the giant dipole and giant quadrupole resonances, integrated cross sections and their moments, nuclear symmetry energies, and nuclear deformation parameters and intrinsic quadrupole moments. No fewer than eight kinds of evidence point to a sudden change of behavior between $^{189}\mathrm{Os}$ and $^{190}\mathrm{Os}$, which could be interpreted as a phase transition from a statically deformed prolate nucleus to a $\ensuremath{\gamma}$-unstable one, in general (but not detailed) agreement with the prediction of a dynamic-collective-model calculation.