Neural oscillations dominate electrophysiological measures of macroscopic brain activity and fluctuations in these rhythms offer an insightful window on cortical excitation, inhibition, and connectivity. However, in recent years the 'classical' picture of smoothly varying oscillations has been challenged by the idea that many 'oscillations' may actually be formed from the recurrence of punctate high-amplitude bursts in activity, whose spectral composition intersects the traditionally defined frequency ranges (e.g. alpha/beta band). This finding offers a new interpretation of measurable brain activity, however neither the methodological means to detect bursts, nor their link to other findings (e.g. connectivity) have been settled. Here, we use a new approach to detect bursts in magnetoencephalography (MEG) data. We show that a time-delay embedded Hidden Markov Model (HMM) can be used to delineate single-region bursts which are in agreement with existing techniques. However, unlike existing techniques, the HMM looks for specific spectral patterns in timecourse data. We characterise the distribution of burst duration, frequency of occurrence and amplitude across the cortex in resting state MEG data. During a motor task we show how the movement related beta decrease and post movement beta rebound are driven by changes in burst occurrence. Finally, we show that the beta band functional connectome can be derived using a simple measure of burst overlap, and that coincident bursts in separate regions correspond to a period of heightened coherence. In summary, this paper offers a new methodology for burst identification and connectivity analysis which will be important for future investigations of neural oscillations.