At present, the Synthetic Aperture Radar (SAR) image classification method based on convolution neural network (CNN) has faced some problems such as poor noise resistance and generalization ability. Spiking neural network (SNN) is one of the core components of brain-like intelligence and has good application prospects. This article constructs a complete SAR image classifier based on unsupervised and supervised learning of SNN by using spike sequences with complex spatio-temporal information. We firstly expound the spiking neuron model, the receptive field of SNN, and the construction of spike sequence. Then we put forward an unsupervised learning algorithm based on STDP and a supervised learning algorithm based on gradient descent. The average classification accuracy of single layer and bilayer unsupervised learning SNN in three categories images on MSTAR dataset is 80.8\% and 85.1\%, respectively. Furthermore, the convergent output spike sequences of unsupervised learning can be used as teaching signals. Based on the TensorFlow framework, a single layer supervised learning SNN is built from the bottom, and the classification accuracy reaches 90.05\%. By comparing noise resistance and model parameters between SNNs and CNNs, the effectiveness and outstanding advantages of SNN are verified. Code to reproduce our experiments is available at \url{this https URL}.