Current deep learning methods for anomaly detection in text rely on supervisory signals in inliers that may be unobtainable or bespoke architectures that are difficult to tune. We study a simpler alternative: fine-tuning Transformers on the inlier data with self-supervised objectives and using the losses as an anomaly score. Overall, the self-supervision approach outperforms other methods under various anomaly detection scenarios, improving the AUROC score on semantic anomalies by 11.6% and on syntactic anomalies by 22.8% on average. Additionally, the optimal objective and resultant learnt representation depend on the type of downstream anomaly. The separability of anomalies and inliers signals that a representation is more effective for detecting semantic anomalies, whilst the presence of narrow feature directions signals a representation that is effective for detecting syntactic anomalies.