Formation of ripples on a supported graphene sheet involves interfacial interaction with the substrate. In this work, graphene was grown on a copper foil by chemical vapor deposition from methane. On thermal quenching from elevated temperatures, we observed the formation of ripples in grown graphene, developing a peculiar topographic pattern in the form of wavy grooves and single/double rolls, roughly honeycomb cells, or their combinations. Studies on pure copper foil under corresponding conditions but without the presence of hydrocarbon revealed the appearance of peculiar patterns on the foil surface, such as dendritic structures that are distinctive not of equilibrium solidified phases but arise from planar and/or convective instabilities driven by solutal and thermal capillary forces. We propose a new origin for the formation of ripples in the course of graphene growth at elevated temperatures, where the topographic pattern formation is governed by dynamic instabilities on the interface of a carbon-catalyst binary system. These non-equilibrium processes can be described based on Mullins-Sekerka and Benard-Marangoni instabilities in diluted binary alloys, which offer control over the ripple texturing through synthesis parameters such as temperature, imposed temperature gradient, quenching rate, diffusion coefficients of carbon in the metal catalyst, and the miscibility gap of the metal catalyst-carbon system.