A dialogue response is malevolent if it is grounded in negative emotions, inappropriate behavior, or an unethical value basis in terms of content and dialogue acts. The detection of malevolent dialogue responses is attracting growing interest. Current research on detecting dialogue malevolence has limitations in terms of datasets and methods. First, available dialogue datasets related to malevolence are labeled with a single category, but in practice assigning a single category to each utterance may not be appropriate as some malevolent utterances belong to multiple labels. Second, current methods for detecting dialogue malevolence neglect label correlation. Therefore, we propose the task of multi-label dialogue malevolence detection and crowdsource a multi-label dataset, multi-label dialogue malevolence detection (MDMD) for evaluation. We also propose a multi-label malevolence detection model, multi-faceted label correlation enhanced CRF (MCRF), with two label correlation mechanisms, label correlation in taxonomy (LCT) and label correlation in context (LCC). Experiments on MDMD show that our method outperforms the best performing baseline by a large margin, i.e., 16.1%, 11.9%, 12.0%, and 6.1% on precision, recall, F1, and Jaccard score, respectively.