Weakly dispersive Boussinesq-type models are extensively used to model long-wave propagation in coastal areas and their interaction with coastal infrastructures. Many equations falling in this category have been formulated during the last decades, but few detailed comparisons between them can be found in the literature. In this work, we investigate theoretically and with computational experiments eight variants of the most popular models used by the coastal engineering community. Both weakly nonlinear and fully nonlinear models are considered, hoping to understand better when the additional complexity of the latter class of models is necessary or justified. We provide an overview and discuss the properties of these models, including the linear dispersion relation in uniform water depth, the second-order nonlinear coupling coefficient, the shoaling gradient, and the sensitivity to wave trough instabilities. The models are then numerically discretised using the same general strategy in a single numerical code, using fourth-order methods for time and space discretisation. Their capacity to simulate coastal wave propagation and their transformation when approaching the shore is assessed on three challenging one-dimensional benchmarks. It appears that fully nonlinear models are more consistent than their weakly nonlinear counterparts, which can occasionally perform better but show different behaviours depending on the case.