Equilibrium conditions and spin dynamics of spinning test particles are discussed in the stationary and axially symmetric Kerr--de~Sitter black-hole or naked-singularity spacetimes. The general equilibrium conditions are established, but due to their great complexity, the detailed discussion of the equilibrium conditions and spin dynamics is presented only in the simple and most relevant case of equilibrium positions in the equatorial plane of the spacetimes. It is shown that due to the combined effectof the rotation of the source and the cosmic repulsion the equilibrium is spin dependent in contrast to the spherically symmetric spacetimes.