The Internet of Things (IoT) is reshaping modern society by allowing a decent number of RF devices to connect and share information through RF channels. However, such an open nature also brings obstacles to surveillance. For alleviation, a surveillance oracle, or a cognitive communication entity needs to identify and confirm the appearance of known or unknown signal sources in real-time. In this paper, we provide a deep learning framework for RF signal surveillance. Specifically, we jointly integrate the Deep Neural Networks (DNNs) and Quickest Detection (QD) to form a sequential signal surveillance scheme. We first analyze the latent space characteristic of neural network classification models, and then we leverage the response characteristics of DNN classifiers and propose a novel method to transform existing DNN classifiers into performance-assured binary abnormality detectors. In this way, we seamlessly integrate the DNNs with the parametric quickest detection. Finally, we propose an enhanced Elastic Weight Consolidation (EWC) algorithm with better numerical stability for DNNs in signal surveillance systems to evolve incrementally, we demonstrate that the zero-bias DNN is superior to regular DNN models considering incremental learning and decision fairness. We evaluated the proposed framework using real signal datasets and we believe this framework is helpful in developing a trustworthy IoT ecosystem.