Properties of the free-energy landscape in phase space of a dense hard-sphere system characterized by a discretized free-energy functional of the Ramakrishnan-Yussouff form are investigated numerically. A considerable number of glassy local minima of the free energy are located and the distribution of an appropriately defined `overlap' between minima is calculated. The process of transition from the basin of attraction of a minimum to that of another one is studied using a new `microcanonical' Monte Carlo procedure, leading to a determination of the effective height of free-energy barriers that separate different glassy minima. The general appearance of the free-energy landscape resembles that of a putting green: deep minima separated by a fairly flat structure. The growth of the effective free-energy barriers with increasing density is consistent with the Vogel-Fulcher law, and this growth is primarily driven by an entropic mechanism.