This paper investigates the use of deep convolutional spiking neural networks (SNN) for keyword spotting (KWS) and wakeword detection tasks. The brain-inspired SNN mimic the spike-based information processing of biological neural networks and they can operate on the emerging ultra-low power neuromorphic chips. Unlike conventional artificial neural networks (ANN), SNN process input information asynchronously in an event-driven manner. With temporally sparse input information, this event-driven processing substantially reduces the computational requirements compared to the synchronous computation performed in ANN-based KWS approaches. To explore the effectiveness and computational complexity of SNN on KWS and wakeword detection, we compare the performance and computational costs of spiking fully-connected and convolutional neural networks with ANN counterparts under clean and noisy testing conditions. The results obtained on the Speech Commands and Hey Snips corpora have shown the effectiveness of the convolutional SNN model compared to a conventional CNN with comparable performance on KWS and better performance on the wakeword detection task. With its competitive performance and reduced computational complexity, convolutional SNN models running on energy-efficient neuromorphic hardware offer a low-power and effective solution for mobile KWS applications.