Cells form spatial patterns by coordinating their gene expressions. How a group of mesoscopic numbers (hundreds to thousands) of cells, without pre-existing morphogen gradients and spatial organization, self-organizes spatial patterns remains poorly understood. Of particular importance are dynamic spatial patterns such as spiral waves that perpetually move and transmit information. We developed an open-source software for simulating a field of cells that communicate by secreting any number of molecules. With this software and a theory, we identified all possible “cellular dialogues”—ways of communicating with two diffusing molecules—that yield diverse dynamic spatial patterns. These patterns emerge despite widely varying responses of cells to the molecules, gene-expression noise, spatial arrangements, and cell movements. A three-stage, “order-fluctuate-settle” process forms dynamic spatial patterns: cells form long-lived whirlpools of wavelets that, following erratic dynamics, settle into a dynamic spatial pattern. Our work helps in identifying gene-regulatory networks that underlie dynamic pattern formations. Dang et al. developed a software and a theoretical framework to discover and classify all moving spatial patterns (e.g., waves) that cells can form by secreting two diffusible molecules that control their gene expressions. They identified all gene regulations that the molecules can have for forming moving patterns, which self-organize through a three-stage, “order-fluctuate-settle” dynamic.