Singing voice conversion (SVC) automates song covers by converting a source singing voice from a source singer into a new singing voice with the same lyrics and melody as the source, but sounds like being covered by the target singer of some given target singing voices. However, it raises serious concerns about copyright and civil right infringements. We propose SongBsAb, the first proactive approach to tackle SVC-based illegal song covers. SongBsAb adds perturbations to singing voices before releasing them, so that when they are used, the process of SVC will be interfered, leading to unexpected singing voices. Perturbations are carefully crafted to (1) provide a dual prevention, i.e., preventing the singing voice from being used as the source and target singing voice in SVC, by proposing a gender-transformation loss and a high/low hierarchy multi-target loss, respectively; and (2) be harmless, i.e., no side-effect on the enjoyment of protected songs, by refining a psychoacoustic model-based loss with the backing track as an additional masker, a unique accompanying element for singing voices compared to ordinary speech voices. We also adopt a frame-level interaction reduction-based loss and encoder ensemble to enhance the transferability of SongBsAb to unknown SVC models. We demonstrate the prevention effectiveness, harmlessness, and robustness of SongBsAb on five diverse and promising SVC models, using both English and Chinese datasets, and both objective and human study-based subjective metrics. Our work fosters an emerging research direction for mitigating illegal automated song covers.