Small oscillations of current-carrying string loops around stable equilibrium positions corresponding to minima of the effective potential in the equatorial plane of the Kerr black holes are studied using the perturbation method. In the lowest approximation, two uncoupled harmonic oscillators are obtained that govern the radial and vertical oscillations; the higher-order terms determine nonlinear phenomena and transition to chaotic motion through quasiperiodic stages of the oscillatory motion. The radial profiles of frequencies of the radial and vertical harmonic oscillations that are also relevant in the quasiperiodic stages of the oscillatory motion are given, and their properties, independent of the spin of the black holes and the angular momentum and tension of the string loops, are determined. It is shown that the radial profiles differ substantially from those corresponding to the radial and vertical frequencies of the geodetical epicyclic motion; however, they have the same mass