We investigate the open-shell systems of the methyl radical, the allyl radical and molecular oxygen using Monte Carlo configuration interaction. We look at whether modifying Monte Carlo configuration interaction to use the increased flexibility of unrestricted Hartree-Fock orbitals offers any benefits in the description of these systems by the resulting compact wavefunctions. The expectation of the total spin squared when using Slater determinants is calculated to investigate if pure spin states can be evolved by Monte Carlo configuration interaction and how this might be accelerated. We consider the multireference character of these open-shell systems. To this end we demonstrate how a previously introduced way [J. P. Coe and M. J. Paterson, J. Chem. Theory Comput. 11, 4189 (2015)] of viewing a multireference indicator of P.-O. Löwdin [P.-O. Löwdin, Phys. Rev. 97, 1474 (1955)] in terms of the spatial entanglement can be generalized to open-shell systems.