  We study the main properties of general linear perturbations of rotating
black holes in asymptotically flat higher-dimensional spacetimes. In
particular, we determine the quasinormal mode (QNM) spectrum of singly spinning
and equal angular momenta Myers-Perry black holes (MP BHs). Emphasis is also
given to the timescale of the ultraspinning and bar-mode instabilities in these
two families of MP BHs. For the bar-mode instabilities in the singly spinning
MP BH, we find excellent agreement with our linear analysis and the non-linear
time evolution of Shibata and Yoshino for d=6,7 spacetime dimensions. We find
that d=5 singly spinning BHs are linearly stable. In the context of studying
general relativity in the large dimension limit, we obtain the QNM spectrum of
Schwarzschild BHs and rotating MP BHs for large dimensions. We identify two
classes of modes. For large dimensions, we find that in the limit of zero
rotation, unstable modes of the MP BHs connect to a class of Schwarzschild QNMs
that saturate to finite values.
