The cellular actomyosin network plays an important role in a wide range of cell behavior, including motility, morphology and mechanotransduction. Although the building blocks of actomyosin networks are well understood at the molecular level, the picture connecting these microscale units to cell-scale behavior remains incomplete. Here we present results using active micropost array detectors (AMPADS) to characterize the dynamical fluctuations and local rheology of cellular actomyosin networks in detail. AMPADS are poly(dimethylsiloxane) (PDMS) micropillar arrays with embedded magnetic nanowires that enable mechanical actuation of an adherent cell. We classified the cell-associated microposts based on their traction force, and identified two distinct populations, one containing posts coupled to stress fibers and the other containg posts coupled to the actomyosin cortex. Both groups show weak power law rheology, but we find that the fluctuations of stress fiber-associated posts are more active and highly anisotropic in comparison to the cortically-associated posts. Notably, the fluctuations of both populations are highly heterogenous and resemble a Lévy process. Specifically, the mean-squared displacements of both types of posts display broadly distributed amplitudes and super-diffusive behavior. We find that the amplitude distribution is fat-tailed, and is dominated by a corresponding distribution of intermittent, large step-like displacements, resembling avalanches and earthquakes in physical systems. Our regular array of detectors also allows us to determine the spatial extent and average symmetry of the largest rearrangements in the cortex, which are both spatially and temporally complex, resembling those in plastic solids. These results suggest that the dynamics of the cellular actomyosin network resembles what is seen in soft matter systems such as sandpiles and foams, where constituent components self-organize into marginally stable, plastic networks, with significant implications for future models of the cortex.