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Accurate online classification of disturbance events in a transmission network is an important part of wide-area monitoring. Although many conventional machine learning techniques are very successful in classifying events, they rely on extracting information from PMU data at control centers and processing them through CPU/GPUs, which are highly inefficient in terms of energy consumption. To solve this challenge without compromising accuracy, this article presents a novel methodology based on event-driven neuromorphic computing architecture for classification of power system disturbances. A Spiking Neural Network (SNN)-based computing framework is proposed, which exploits sparsity in disturbances and promotes local event-driven operation for unsupervised learning and inference from incoming data. Spatio-temporal information of PMU signals is first extracted and encoded into spike trains and classification is achieved with SNN-based supervised and unsupervised learning framework. In addition, benefits of deep spiking networks for complex multi-class event identification problem are presented by leveraging increasing dynamic neural sparse spiking events with network depth. Moreover, a QR decomposition-based selection technique is proposed to identify signals participating in the low rank subspace of multiple disturbance events. Performance of the proposed method is validated on data collected from a 16-machine, 5-area New England-New York system.