Time series of brain activity recorded from different anatomical regions and in different behavioural states and pathologies can be summarised by the power spectrum. Recently, attention has shifted to characterising the properties of changing temporal dynamics in rhythmic neural activity. Here, we present evidence from electrocorticography recordings made from the motor cortex to show that, dependent on the specific motor context, the statistics of temporal transients in beta frequency (14-30 Hz) rhythms (i.e., bursts) can significantly add to the description of states such rest, movement preparation, movement execution, and movement imagery. We show that the statistics of burst duration and amplitude can significantly improve the classification of motor states and that burst features reflect nonlinearities not detectable in the power spectrum, with states increasing in order of nonlinearity from movement execution to movement preparation to rest. Further, we provide mechanistic explanations for these features by fitting models of the motor cortical microcircuit to the empirical data and investigate how dynamical instabilities interact with noise to generate burst dynamics. Finally, we examine how beta bursting in motor cortex may influence the integration of exogenous inputs to the cortex and suggest that properties of spontaneous activity cannot be reliably used to infer the response of the cortex to external inputs. These findings have significance for the classification of motor states, for instance in novel brain-computer interfaces. Critically, we increase the understanding of how transient brain rhythms may contribute to cortical processing, which in turn, may inform novel approaches for its modulation with brain stimulation.