We study the spin polarization of the ground states and the excited states of the fractional quantum Hall effect, using spherical geometry for finite-size systems. In the absence of the Zeeman energy, the ground states at filling 2/q, with q odd, are found to be spin unpolarized and nondegenerate for all values of q studied. The energy differences between the spin-polarized and spin-unpolarized states at \ensuremath{\nu}=2/3 and 2/5 are estimated by extrapolating to the thermodynamic limit. The spin polarization of the lowest-energy excitations are studied and the results agree with recent experiments.