Our recent experimental studies on the 1/8 anomaly in the Bi-2212 and Y-123 phases and on the new anomaly at x = 0.21 in La2-xSr x Cuiy ZnyO4 are reviewed. In the Zn-substituted Bi-2212 and Y-123 phases, we have found anomalous suppression of superconductivity at p (the hole concentration per Cu in the CuCh plane)-1/8, and it has been found from the muon spin relaxation (// SR) measurements that the magnetic correlation between Cu spins is enhanced singularly at p ~ 1/8. Furthermore, we have found marked suppression of superconductivity singularly at x = 0.21 in the 1% Zn-substituted single-crystals of La2-xSr x Cuiy ZnyO4, and it has been found from the // SR measurements that the magnetic correlation is also enhanced at x = 0.21. These results suggest that the dynamical stripe correlations of holes and spins exist in the Bi-2212 and Y-123 phases as well as in the La-214 phase and that they tend to be pinned by Zn, leading to the suppression of superconductivity. If this tendency is the case in a wide range of p, it is possible to understand the universal results in the high-T c cuprates that Zn is a strong scatterer and that the Zn substitution markedly suppresses superconductivity.