In recent years, sparse spectral methods for solving partial differential equationshave been derived using hierarchies of classical orthogonal polynomials on intervals,disks, disk-slices and triangles. In this work we extend the methodology to a hierarchyof non-classical multivariate orthogonal polynomials on spherical caps. The entries ofdiscretisations of partial differential operators can be effectively computed using for-mulae in terms of (non-classical) univariate orthogonal polynomials. We demonstratethe results on partial differential equations involving the spherical Laplacian and bihar-monic operators, showing spectral convergence with discretisations that can be madewell-conditioned using a simple preconditioner.