Cellular signaling systems precisely transmit information in the presence of molecular noise while retaining flexibility to accommodate the needs of individual cells. To understand design principles underlying such versatile signaling, we analyzed the response of the tumor suppressor p53 to varying levels of DNA damage in hundreds of individual cells and observed a switch between distinct signaling modes characterized by isolated pulses and sustained oscillations of p53 accumulation. Guided by dynamic systems theory we show that this requires an excitable network structure comprising positive feedback and provide experimental evidence for its molecular identity. The resulting dataN driven model reproduced all features of measured signaling responses and explained their heterogeneity in individual cells. We predicted and validated that heterogeneity in the levels of the feedback regulator Wip1 sets cellNspecific thresholds for p53 activation, providing means to modulate its response through interacting signaling pathways. Our results demonstrate how excitable signaling networks provide high specificity, sensitivity and robustness while retaining unique possibilities to adjust their function to the physiology of individual cells.