When the spins of molecular orbitals are allowed to be aligned with different directions in space rather than being aligned collinearly, the resulting noncollinear spin orbitals add extra flexibility to variational optimization of the orbitals, and solutions obtained with collinear spin orbitals may be unstable with respect to becoming noncollinear in the expanded variational space. The goal of the present work is to explore whether and in what way the molecular orbitals of the Kohn-Sham density functional theory become noncollinear when fully optimized for multi-reference molecules, transition states, and reaction paths. (We note that a noncollinear determinant has intermediate flexibility between a collinear determinant and a linear combination of many collinear determinants with completely independent coefficients. However, the Kohn-Sham method is defined to involve the variational optimization of a single determinant, and a noncollinear determinant represents the limit of complete optimization in the Kohn-Sham scheme.) We compare the results obtained with the noncollinear Kohn-Sham (NKS) scheme to those obtained with the widely used unrestricted Kohn-Sham (UKS) scheme for two types of multi-reference systems. For the dissociation of the MnO and NiO transition metal oxides, we find UKS fails to dissociate to the ground states of neutral atoms, while NKS dissociates to the correct limit and predicts potential energy curves that vary smoothly at intermediate bond lengths. This is due to the instability of UKS solutions at large bond distances. For barrier heights of O3, BeH2, and H4, NKS is shown to stabilize the multi-reference transition states by expanding the variational space. Although the errors vary because they are closely coupled with the capability of the employed exchange-correlation functionals in treating the multi-configurational states, these findings demonstrate that results with collinear spin orbitals should be further scrutinized, and future development of exchange-correlation functionals for multi-reference systems should incorporate the flexibilities of NKS.