We consider a mean-field density-functional model for three-phase equilibria and wetting. The model features two densities and two control parameters, one of which is related to order parameter asymmetry or spatial anisotropy. The global wetting phase diagram in the space of these two parameters is rich. It features first-order, second-order, continuously-varying-order and infinite-order wetting transitions. The divergence of the wetting layer thickness is usually logarithmic as a function of the distance to the transition, but, in contrast, algebraic upon approach of an infinite-order transition. Further, an approximate interface potential approach is proposed, which allows us to derive analytic predictions for the singular behaviour of thermodynamic functions near wetting, in accordance with accurate numerical computations. It is conjectured that previously developed mean-field models, such as, for example, one for ferromagnets with cubic anisotropy also contain a segment of infinite-order transitions. We speculate that the possibility of varying the spatial anisotropy of the magnetic interaction in these systems might well lead the way towards the first experimental realization of infinite-order wetting.