The development of any atmosphere or ocean model warrants a suite of test cases to verify its spatial and temporal discretizations, order of accuracy, stability, reproducability, portability, scalability, etc. In this paper, we present a suite of shallow water test cases designed to verify the barotropic solver of atmosphere and ocean models. These include the non-dispersive coastal Kelvin wave; the dispersive inertia-gravity wave; the dispersive planetary and topographic Rossby waves; the barotropic tide; and a non-linear manufactured solution. These test cases check the implementation of the linear pressure gradient term; the linear constant or variable-coefficient Coriolis and bathymetry terms; and the non-linear advection terms. Simulation results are presented for a variety of time-stepping methods as well as two spatial discretizations: a mimetic finite volume method based on the TRiSK scheme, and a high-order discontinuous Galerkin spectral element method. We explain the strategies that need to be adopted for specifying initial and non-periodic boundary conditions on hexagonal meshes. Convergence studies of every test case are conducted with refinement in both space and time, only in space, and only in time. The convergence slopes match the expected theoretical predictions.