We utilize a simple three-variable reaction-diffusion model to study patterns that emerge beyond the onset of the (short-)wave instability. We have found various wave patterns including standing waves, traveling waves, asymmetric standing-traveling waves and target patterns. We employ both periodic and zero flux boundary conditions in the simulations, and we analyze the patterns using space-time two-dimensional Fourier spectra. A fascinating pattern of waves which periodically change their direction of propagation along a ring is found for very short systems. A related pattern of modulated standing waves is found for systems with zero flux boundary conditions. In a two-dimensional system with small overcriticality we observe a wide variety of standing wave patterns. These include plain and modulated stripes, squares and rhombi. We also find standing waves consisting of periodic time sequences of stripes, rhombi and hexagons. The short-wave instability can lead to a much greater variety of spatio-temporal patterns than the aperiodic Turing and the long-wave oscillatory instabilities. For example, a single oscillatory cycle may display all the basic patterns related to the aperiodic Turing instability - stripes, hexagons and inverted hexagons (honeycomb) - as well as rhombi and modulated stripes. A rich plethora of patterns is seen in a system with cylindrical geometry - examples include rotating patterns of standing waves and counter-propagating waves.