High-spin states of the odd-neutron 121Xe nucleus have been studied with Eurogam using the 96Zr(30Si,5n)121Xe fusion-evaporation reaction. The level scheme has been extended up to 67/2h(cross) and an excitation energy of approximately 14 MeV. Several new rotational bands have been observed and the previously known bands extended. Two of the bands lose their regular character at high spins, which may be interpreted as a transition from collective behaviour to a regime of noncollective oblate states. The deduced high-spin structure is compared with Woods-Saxon TRS cranking and CSM calculations. Configurations of the bands have been suggested on the basis of the measured routhians, aligned angular momenta and B(M1)/B(E2) ratios. The yrast nu h(cross)11/2 band is interpreted as having a triaxial shape. Neutron (h(cross)11/2)2 alignments have been found at the frequencies predicted by the TRS and CSM calculations. The B(M1)/B(E2) ratios for one of the bands show deviation from the predictions of the geometrical model. Enhanced E1 transitions have been observed between bands built on nu d5/2 and nu h(cross)11/2 orbitals.