Camera model identification (CMI) can be widely used in image forensics such as authenticity determination, copyright protection, forgery detection, etc. Meanwhile, with the vigorous development of the Internet, online social networks (OSNs) have become the dominant channels for image sharing and transmission. However, the inevitable lossy operations on OSNs, such as compression and post-processing, impose great challenges to the existing CMI schemes, as they severely destroy the camera traces left in the images under investigation. In this work, we propose a novel CMI method that is robust against the lossy operations of various OSN platforms. Specifically, it is observed that a camera trace extractor can be easily trained on a single degradation scenario (e.g., one specific OSN platform); while much more difficult on mixed degradation scenarios (e.g., multiple OSN platforms). Inspired by this observation, we design a new multi-scenario learning (MSL) strategy, enabling us to extract robust camera traces across different OSNs. Furthermore, noticing that image smooth regions incur less distortions by OSN and less interference by image signal itself, we suggest a SmooThness-Aware Trace Extractor (STATE) that can adaptively extract camera traces according to the smoothness of the input image. The superiority of our method is verified by comparative experiments with four state-of-the-art methods, especially under various OSN transmission scenarios. Particularly, for the open-set camera model verification task, we greatly surpass the second-place by 15.30% in AUC on the FODB dataset; while for the close-set camera model classification task, we are significantly ahead of the second-place by 34.51% in F1 on the SIHDR dataset. The code of our proposed method is available at https://github.com/HighwayWu/CameraTraceOSN.