Nonequilibrium spin distributions in single GaAs/AlGaAs core-shell nanowires are excited using resonant polarized excitation at 10 K. At all excitation energies, we observe strong photoluminescence polarization due to suppressed radiative recombination of excitons with dipoles aligned perpendicular to the nanowire. Excitation resonances are observed at 1- or 2-LO phonon energies above the exciton ground states. Using rate equation modeling, we show that, at the lowest energies, strongly nonequilibrium spin distributions are present and we estimate their spin relaxation rate.