Cyberbullying on social media is inherently multilingual and multi-faceted, where abusive behaviors often overlap across multiple categories. Existing methods are commonly limited by monolingual assumptions or single-task formulations, which restrict their effectiveness in realistic multilingual and multi-label scenarios. In this paper, we propose HMS-BERT, a hybrid multi-task self-training framework for multilingual and multi-label cyberbullying detection. Built upon a pretrained multilingual BERT backbone, HMS-BERT integrates contextual representations with handcrafted linguistic features and jointly optimizes a fine-grained multi-label abuse classification task and a three-class main classification task. To address labeled data scarcity in low-resource languages, an iterative self-training strategy with confidence-based pseudo-labeling is introduced to facilitate cross-lingual knowledge transfer. Experiments on four public datasets demonstrate that HMS-BERT achieves strong performance, attaining a macro F1-score of up to 0.9847 on the multi-label task and an accuracy of 0.6775 on the main classification task. Ablation studies further verify the effectiveness of the proposed components.