This letter deals with a study of aperiodically ordered textured (mushroom-type) high-impedance surfaces (HISs), and their possible application as artificial-magnetic-conductor ground-planes for low-profile directive antennas. In this framework, results from full-wave simulations are presented in order to characterize the electromagnetic response (in terms of impedance matching, directivity, and surface-wave suppression) of selected configurations based on representative categories of aperiodic tilings, and compare them with those of standard periodic HISs.