The evolutions of spinning test bodies are investigated in rotating (Kerr, Bardeen-like and Hayward-like) black hole spacetimes. Spin vector precessional equations are derived in both comoving and zero 3-momentum frames from the Mathisson-Papapetrou-Dixon (MPD) equations using either the Frenkel-Mathisson-Pirani or the Tulczyjew-Dixon spin supplementary condition. The comoving and the zero 3-momentum frames are set up from either the static or the zero angular momentum observer frame by instantaneous Lorentz-boosts. However when the body passes over the ergosphere only the boosted zero angular momentum frame can be used for description of the spin dynamics during the whole evolution. Far from the black hole the difference between the boosted static and zero angular momentum frames is unsignificant. Numerical applications are presented for spinning bodies moving along spherical-like, zoom-whirl (thus their existence is confirmed based on the MDP equations) and unbound orbits. The spin evolutions are presented in the boosted static and zero angular momentum observer frames and they are compared, obtaining only differences in the near black hole region. We have found the spin magnitude influences on the orbit evolution, and the spin precessional angular velocity is highly increased near and inside the ergosphere.