We propose a snapshot compressive structured illumination microscopy (SoSIM) system to increase the number of reconstructed resolution-enhanced (RE) images per second and reduce the data bandwidth by capturing compressed measurements. In this system, multiple low-resolution images are encoded by a high-speed digital micro-mirror device with random binary masks. These images are then captured by a low-speed camera as a snapshot compressed measurement. Following this, we adopt an efficient deep neural network to reconstruct nine images with different structured illumination patterns from a single measurement. The reconstructed images are then combined into a single-frame RE image using the method of spectral synthesis in the frequency domain. When the camera operates at 100 frames per second (fps), we can eventually recover dynamic RE videos at the same speed with 100 fps.