Various patterns of standing waves are found beyond the onset of the short-wave instability in a model reaction-diffusion system. These include plain and modulated stripes, squares, and rhombi in systems with square and rectangular geometry and patterns with rotational symmetry in systems with circular geometry. We also find standing waves consisting of periodic time sequences of stripes and rhombi, stripes and squares, and 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 instance, a single oscillatory cycle can display all the basic patterns related to the aperiodic Turing instability-stripes, hexagons, and inverted hexagons (honeycomb)-as well as rhombi and modulated stripes.