The rapid pace of extrasolar planet discovery and characterization is legitimizing the study of their atmospheres via three-dimensional numerical simulations. The complexity of atmospheric modelling and its inherent non-linearity, together with the limited amount of data available, motivate model intercomparisons and benchmark tests. In the geophysical community, the Held–Suarez test is a standard benchmark for comparing dynamical core simulations of the Earth’s atmosphere with different solvers, based on statistically averaged flow quantities. In the present study, we perform analogues of the Held–Suarez test for tidally locked exoplanets with the Geophysical Fluid Dynamics Laboratory (GFDL) Princeton Flexible Modelling System (fms) by subjecting both the spectral and finite difference dynamical cores to a suite of tests, including the standard benchmark for the Earth, a hypothetical tidally locked Earth, a ‘shallow’ hot Jupiter model and a ‘deep’ model of HD 209458b. We find qualitative and quantitative agreement between the solvers for the Earth, tidally locked Earth and shallow hot Jupiter benchmarks, but the agreement is less than satisfactory for the deep model of HD 209458b. Further investigation reveals that closer agreement may be attained by arbitrarily adjusting the values of the horizontal dissipation parameters in the two solvers, but it remains the case that the magnitude of the horizontal dissipation is not easily specified from first principles. Irrespective of radiative transfer or chemical composition considerations, our study points to limitations in our ability to accurately model hot Jupiter atmospheres with meteorological solvers at the level of 10 per cent for the temperature field and several tens of per cent for the velocity field. Direct wind measurements should thus be particularly constraining for the models. Our suite of benchmark tests also provides a reference point for researchers wishing to adapt their codes to study the atmospheric circulation regimes of tidally locked Earths/Neptunes/Jupiters.