Music plagiarism detection is gaining more and more attention due to the popularity of music production and society's emphasis on intellectual property. We aim to find fine-grained plagiarism in music pairs since conventional methods are coarse-grained and cannot match real-life scenarios. Considering that there is no sizeable dataset designed for the music plagiarism task, we establish a large-scale simulated dataset, named Music Plagiarism Detection Dataset (MPD-Set) under the guidance and expertise of renowned researchers from national-level professional institutions in the field of music. MPD-Set considers diverse music plagiarism cases found in real life from the melodic, rhythmic, and tonal levels respectively. Further, we establish a Real-life Dataset for evaluation, where all plagiarism pairs are real cases. To detect the fine-grained plagiarism pairs effectively, we propose a graph-based method called Bipatite Melody Matching Detector (BMM-Det), which formulates the problem as a max matching problem in the bipartite graph. Experimental results on both the simulated and Real-life Datasets demonstrate that BMM-Det outperforms the existing plagiarism detection methods, and is robust to common plagiarism cases like transpositions, pitch shifts, duration variance, and melody change. Datasets and source code are open-sourced at https://github.com/xuan301/BMMDet_MPDSet.