Large loss spikes in stochastic gradient descent are studied through a rigorous large-deviations analysis for a shallow, fully connected network in the NTK scaling. In contrast to full-batch gradient descent, the catapult phase is shown to split into inflationary and deflationary regimes, determined by an explicit log-drift criterion. In both cases, large spikes are shown to be at least polynomially likely. In addition, these spikes are shown to be the dominant mechanism by which sharp minima are escaped and curvature is reduced, thereby favouring flatter solutions. Corresponding results are also obtained for certain ReLU networks, and implications for curriculum learning are derived.