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Future communication networks must address the scarce spectrum to accommodate extensive growth of heterogeneous wireless devices. Wireless signal recognition is becoming increasingly more significant for spectrum monitoring, spectrum management, secure communications, among others. Consequently, comprehensive spectrum awareness on the edge has the potential to serve as a key enabler for the emerging beyond 5G networks. State-of-the-art studies in this domain have (i) only focused on a single task - modulation or signal (protocol) classification - which in many cases is insufficient information for a system to act on, (ii) consider either radar or communication waveforms (homogeneous waveform category), and (iii) does not address edge deployment during neural network design phase. In this work, for the first time in the wireless communication domain, we exploit the potential of deep neural networks based multi-task learning (MTL) framework to simultaneously learn modulation and signal classification tasks while considering heterogeneous wireless signals such as radar and communication waveforms in the electromagnetic spectrum. The proposed MTL architecture benefits from the mutual relation between the two tasks in improving the classification accuracy as well as the learning efficiency with a lightweight neural network model. We additionally include experimental evaluations of the model with over-the-air collected samples and demonstrate first-hand insight on model compression along with deep learning pipeline for deployment on resource-constrained edge devices. We demonstrate significant computational, memory, and accuracy improvement of the proposed model over two reference architectures. In addition to modeling a lightweight MTL model suitable for resource-constrained embedded radio platforms, we provide a comprehensive heterogeneous wireless signals dataset for public use.