We present accurate optimised-potential-model (OPM) results for spherical spin-polarised atoms emphasising the precise construction of the OPM exchange potential from the numerical solution of the OPM integral equation, especially for large r. The results are used to discuss the quality of the local spin-density approximation (LSDA) and a generalised-gradient expansion (GGA) [A. D. Becke, Phys. Rev. A 98, 3098 (1988)] for describing these atoms. It is shown that the LSDA can produce substantial errors (beyond what is known from unpolarised atoms) for quantities which are directly related to the spin polarisation of these systems. In particular, the LSDA overestimates the magnetisation density in the interior of Cu by a factor of 2