At the mesoscale, dislocation activity is known to be heterogeneous and intermittent and to proceed in avalanche-like plastic bursts. While material hardening may well be understood in terms of average dislocation behaviour, other aspects such as strain rate sensitivity, plastic localization and hence stress concentrations are intimately linked to dislocation avalanches. Many aspects of the critical nature of plastic deformation remain to be understood, and most of existing mesoscale studies, while insightful, concerned 2D geometries or focus on finite-size geometries such as micropillars. In this work, we propose a detailed analysis of dislocation avalanche statistics occurring in bulk Cu single crystals, as described in carefully designed three-dimensional Dislocation Dynamics Simulations. Large data sets reveal a well-defined critical behaviour associated to a Power Law (PWL) that extends over 3 to 5 decades of plastic event sizes. The well defined cut-offs delimitating the PWL regimes are qualitatively correlated with the properties of the dislocation microstructure. The simulated dislocation microstructures are thoroughly analysed to reveal critical configurations and associated distribution at the origin of the dislocation avalanches. Simulations results on avalanche statistics provide unique insights into the role of individual slip systems, cross-slip events or competition with the applied strain rate. These simulation results help to understand experimental results, such as in-house accoustic emission data, and can provide mechanistic explanations of the experimental avalanche signature.