Curvelets are efficient to represent highly anisotropic signal content, such as a local linear and curvilinear structure. First-generation curvelets on the sphere, however, suffered from blocking artefacts. We present a new second-generation curvelet transform, where scale-discretised curvelets are constructed directly on the sphere. Scale-discretised curvelets exhibit a parabolic scaling relation, are well-localised in both spatial and harmonic domains, support the exact analysis and synthesis of both scalar and spin signals, and are free of blocking artefacts. We present fast algorithms to compute the exact curvelet transform, reducing computational complexity from $\mathcal{O}(L^5)$ to $\mathcal{O}(L^3\log_{2}{L})$ for signals band-limited at $L$. The implementation of these algorithms is made publicly available. Finally, we present an illustrative application demonstrating the effectiveness of curvelets for representing directional curve-like features in natural spherical images.