Nowadays, mobile devices enable constant tracking of the user's position and location trajectories can be used to infer personal points of interest (POIs) like homes, workplaces, or stores. A common way to extract POIs is to first identify spatio-temporal regions where a user spends a significant amount of time, known as stay regions (SRs). Common approaches to SR extraction are evaluated either solely unsupervised or on a small-scale private dataset, as popular public datasets are unlabeled. Most of these methods rely on hand-crafted features or thresholds and do not learn beyond hyperparameter optimization. Therefore, we propose a weakly and self-supervised transformer-based model called DeepStay, which is trained on location trajectories to predict stay regions. To the best of our knowledge, this is the first approach based on deep learning and the first approach that is evaluated on a public, labeled dataset. Our SR extraction method outperforms state-of-the-art methods. In addition, we conducted a limited experiment on the task of transportation mode detection from GPS trajectories using the same architecture and achieved significantly higher scores than the state-of-the-art. Our code is available at this https URL