Front solutions of the one-dimensional Richards’ equation used to describe groundwater flow are studied systematically for the three soil retention models known as Brooks–Corey, Mualem–Van Genuchten and Storm–Fujita. Both the infiltration problem when water percolates from the surface into the ground under the influence of gravity and the imbibition (absorption) problem when groundwater diffuses in the horizontal direction without the gravity effect are considered. In general, self-similar solutions of the first kind in the form of front exist only for the imbibition case; such solutions are stable against small perturbations. In the particular case of the Brooks–Corey model, self-similar solutions of the second kind in the form of a decaying pulse also exist both for the imbibition and infiltration cases. Steady-state solutions in the form of travelling fronts exist for the infiltration case only. The existence of such solutions does not depend on the specifics of the soil retention model. It is shown numerically that these solutions are stable against small perturbations.