Microscopy is an essential tool in a huge range of research areas. Until now, microscopy has been largely restricted to imaging in the visible region of the electromagnetic spectrum. Here we present a microscope system that uses single-pixel imaging techniques to produce images simultaneously in the visible and shortwave infrared. We apply our microscope to the inspection of various objects, including a silicon CMOS sensor, highlighting the complementarity of the visible and shortwave infrared wavebands. The system is capable of producing images with resolutions between 32×3232×32 and 128×128128×128 pixels at corresponding frame rates between 10 and 0.6 Hz. We introduce a compressive technique that does not require postprocessing, resulting in a predicted frame rate increase by a factor 8 from a compressive ratio of 12.5% with only 28% relative error.