Fluctuating oscillations are a ubiquitous feature of neurophysiology. Are the amplitude fluctuations of neural oscillations chance excursions drawn randomly from a normal distribution, or do they tell us more? Recent empirical research suggests that the occurrence of 'anomalous' (high amplitude) oscillations imbues their probability distributions with a heavier tail than the standard normal distribution. However, not all heavy tails are the same. We provide canonical examples of different heavy-tailed distributions in cortical oscillations and discuss the corresponding mechanisms that each suggest, ranging from criticality to multistability, memory, bifurcations, and multiplicative noise. Their existence suggests that the brain is a strongly correlated complex system that employs many different functional mechanisms, and that likewise, we as scientists should refrain from methodological monism.