We demonstrate a fast spin-s spherical harmonic transform algorithm, which is flexible and exact for band-limited functions. In contrast to previous work, where spin transforms are computed independently, our algorithm permits the computation of several distinct spin transforms simultaneously. Specifically, only one set of special functions is computed for transforms of quantities with any spin, namely the Wigner d matrices evaluated at π/2, which may be computed with efficient recursions. For any spin, the computation scales as O(L^3), where L is the band limit of the function. Our publicly available numerical implementation permits very high accuracy at modest computational cost. We discuss applications to the cosmic microwave background and gravitational lensing.