Event-driven acquisition uses neural-network-based recognition of specific biological events to trigger switching between slow and fast super-resolution imaging, enriching the capture of interesting events with high spatiotemporal resolution. A common goal of fluorescence microscopy is to collect data on specific biological events. Yet, the event-specific content that can be collected from a sample is limited, especially for rare or stochastic processes. This is due in part to photobleaching and phototoxicity, which constrain imaging speed and duration. We developed an event-driven acquisition framework, in which neural-network-based recognition of specific biological events triggers real-time control in an instant structured illumination microscope. Our setup adapts acquisitions on-the-fly by switching between a slow imaging rate while detecting the onset of events, and a fast imaging rate during their progression. Thus, we capture mitochondrial and bacterial divisions at imaging rates that match their dynamic timescales, while extending overall imaging durations. Because event-driven acquisition allows the microscope to respond specifically to complex biological events, it acquires data enriched in relevant content.