As artificial neural networks become increasingly integrated into safety-critical systems such as autonomous vehicles, devices for medical diagnosis, and industrial automation, ensuring their reliability in the face of random hardware faults becomes paramount. This paper introduces SpikingJET, a novel fault injector designed specifically for fully connected and convolutional Spiking Neural Networks (SNNs). Our work underscores the critical need to evaluate the resilience of SNNs to hardware faults, considering their growing prominence in real-world applications. SpikingJET provides a comprehensive platform for assessing the resilience of SNNs by inducing errors and injecting faults into critical components such as synaptic weights, neuron model parameters, internal states, and activation functions. This paper demonstrates the effectiveness of Spiking-JET through extensive software-level experiments on various SNN architectures, revealing insights into their vulnerability and resilience to hardware faults. Moreover, highlighting the importance of fault resilience in SNNs contributes to the ongoing effort to enhance the reliability and safety of Neural Network (NN)-powered systems in diverse domains.