Detecting anomalous behavior in wireless spectrum is a demanding task due to the sheer complexity of the electromagnetic spectrum use. Wireless spectrum anomalies can take a wide range of forms from the presence of an unwanted signal in a licensed band to the absence of an expected signal, which makes manual labeling of anomalies difficult and suboptimal. We present, Spectrum Anomaly Detector with Interpretable FEatures (SAIFE), an Adversarial Autoencoder (AAE) based anomaly detector for wireless spectrum anomaly detection using Power Spectral Density (PSD) data which achieves good anomaly detection and localization in an unsupervised setting. In addition, we investigate the model's capabilities to learn interpretable features such as signal bandwidth, class and center frequency in a semi-supervised fashion. Along with anomaly detection the model exhibits promising results for lossy PSD data compression up to 120X and semisupervised signal classification accuracy close to 100% on three datasets just using 20% labeled samples. Finally the model is tested on data from one of the distributed Electrosense sensors over a long term of 500 hours showing its anomaly detection capabilities.