Sparse spectral methods for solving partial differential equations have been derivedin recent years using hierarchies of classical orthogonal polynomials on intervals, disks,and triangles. In this work we extend this methodology to a hierarchy of non-classicalorthogonal polynomials on disk slices and trapeziums. This builds on the observationthat sparsity is guaranteed due to the boundary being defined by an algebraic curve,and that the entries of partial differential operators can be determined using formulaein terms of (non-classical) univariate orthogonal polynomials. We apply the frameworkto solving the Poisson, variable coefficient Helmholtz, and Biharmonic equations. Inthis paper we focus on constant Dirichlet boundary conditions, as well as zero Dirichletand Neumann boundary conditions, with other types of boundary conditions requiringfuture work.