  Complete determination of the polarisation state of light requires at least
four distinct projective measurements of the associated Stokes vector.
Stability of state reconstruction, however, hinges on the condition number
$\kappa$ of the corresponding instrument matrix. Optimisation of redundant
measurement frames with an arbitrary number of analysis states, $m$, is
considered in this Letter in the sense of minimisation of $\kappa$. The minimum
achievable $\kappa$ is analytically found and shown to be independent of $m$,
except for $m=5$ where this minimum is unachievable. Distribution of the
optimal analysis states over the Poincar\'e sphere is found to be described by
spherical 2-designs, including the Platonic solids as special cases. Higher
order polarisation properties also play a key role in nonlinear, stochastic and
quantum processes. Optimal measurement schemes for nonlinear measurands of
degree $t$ are hence also considered and found to correspond to spherical
$2t$-designs, thereby constituting a generalisation of the concept of mutually
unbiased bases.
