We provide an in-depth investigation of quasinormal-mode oscillations of Kerr black holes with nearly extremal angular momenta. We first discuss in greater detail the two distinct types of quasinormal-mode frequencies presented in a recent paper [H. Yang et al., Phys. Rev. D 87, 041502 (2013)]. One set of modes that we call “zero-damping modes” has vanishing imaginary part in the extremal limit and exists for all corotating perturbations (ie modes with azimuthal index m≥ 0). The other set (the “damped modes”) retains a finite decay rate even for extremal Kerr black holes and exists only for a subset of corotating modes. As the angular momentum approaches its extremal value, the frequency spectrum bifurcates into these two distinct branches when both types of modes are present. We discuss the physical reason for the mode branching by developing and using a bound-state formulation for the perturbations of …