The possible exotic cluster decay modes of some ``stable'' nuclei in the region 50Z82 are calculated using a preformed cluster model. The predicted half-life times for certain cluster decays of $_{56}^{120}\mathrm{Ba}_{64}$ and $_{80}^{186}\mathrm{Hg}_{106}$ nuclei, which lie in the neighborhood of the spherical magic shells Z=50 and 82 and the deformed magic shell N=108, are interesting cases of possible instabilities present in regions of ``stable'' nuclei. These are the $^{12}\mathrm{C}$ and $^{16}\mathrm{O}$ decays of $^{120}\mathrm{Ba}$ and the $^{8}\mathrm{Be}$ decay of $^{186}\mathrm{Hg}$, whose half-life times are predicted to be \ensuremath{\sim}${10}^{22}$, ${10}^{26}$, and ${10}^{28}$ s, respectively. The last case is an exciting new possibility of cluster radioactivity with a deformed daughter nucleus. Experimentally, however, both parent nuclei are short lived and their suggested cluster decays cannot be observed. In any case, the present study invokes further work to look for other similar instabilities in various regions of ``stable'' nuclei. It is possible that some of these decays are measurable and can be used to obtain the related new nuclear structure effects. The effects of reinforcement of shell gaps for nuclei with weak spherical closed shell at Z=64 and deformation starting to begin at N\ensuremath{\sim}90 are also discussed.