Medium-voltage (MV) cables often experience a shortened lifespan attributed to insulation breakdown resulting from accelerated aging and anomalous operational and environmental stresses. While partial discharge (PD) measurements serve as valuable tools for assessing the insulation state, complexity arises from the presence of diverse discharge sources, making the evaluation of PD data challenging. The reliability of diagnostics for MV cables hinges on the precise interpretation of PD activity. To streamline the repair and maintenance of cables, it becomes crucial to discern and categorize PD types accurately. This paper presents a comprehensive review encompassing the realms of detection, feature extraction, artificial intelligence, and optimization techniques employed in the classification of PD signals/sources. Its exploration encompasses a variety of sensors utilized for PD detection, data processing methodologies for efficient feature extraction, optimization techniques dedicated to selecting optimal features, and artificial intelligence-based approaches for the classification of PD sources. This synthesized review not only serves as a valuable reference for researchers engaged in the application of methods for PD signal classification but also sheds light on potential avenues for future developments of techniques within the context of MV cables.