  For the representation of spin-$s$ band-limited functions on the sphere, we
propose a sampling scheme with optimal number of samples equal to the number of
degrees of freedom of the function in harmonic space. In comparison to the
existing sampling designs, which require ${\sim}2L^2$ samples for the
representation of spin-$s$ functions band-limited at $L$, the proposed scheme
requires $N_o=L^2-s^2$ samples for the accurate computation of the spin-$s$
spherical harmonic transform~($s$-SHT). For the proposed sampling scheme, we
also develop a method to compute the $s$-SHT. We place the samples in our
design scheme such that the matrices involved in the computation of $s$-SHT are
well-conditioned. We also present a multi-pass $s$-SHT to improve the accuracy
of the transform. We also show the proposed sampling design exhibits superior
geometrical properties compared to existing equiangular and Gauss-Legendre
sampling schemes, and enables accurate computation of the $s$-SHT corroborated
through numerical experiments.
