We have shown that higher dimensional Reissner-Nordstrom-de Sitter black holes are gravitationally unstable for large values of the electric charge and cosmological constant in $D geq 7$ space-time dimensions. We have found the shape of the slightly
perturbed black hole at the threshold point of instability. Why only $D=4, 5$ and 6 dimensional worlds are favorable as to the black stability remains unknown.
The squashed Kaluza-Klien (KK) black holes differ from the Schwarzschild black holes with asymptotic flatness or the black strings even at energies for which the KK modes are not excited yet, so that squashed KK black holes open a window in higher di
mensions. Another important feature is that the squashed KK black holes are apparently stable and, thereby, let us avoid the Gregory-Laflamme instability. In the present paper, the evolution of scalar and gravitational perturbations in time and frequency domains is considered for these squashed KK black holes. The scalar field perturbations are analyzed for general rotating squashed KK black holes. Gravitational perturbations for the so called zero mode are shown to be decayed for non-rotating black holes, in concordance with the stability of the squashed KK black holes. The correlation of quasinormal frequencies with the size of extra dimension is discussed.
