An accelerator-based MeV ultrafast electron microscope (MUEM) has been proposed as a promising tool to the study structural dynamics at the nanometer spatial scale and the picosecond temporal scale. Here, we report experimental tests of a prototype MUEM where high quality images with nanoscale fine structures were recorded with a pulsed ∼3 MeV picosecond electron beam. The temporal and spatial resolutions of the MUEM operating in the single-shot mode are about 4 ps (FWHM) and 100 nm (FWHM), corresponding to a temporal-spatial resolution of 4 × 10−19 s m, about 2 orders of magnitude higher than that achieved with state-of-the-art single-shot keV UEM. Using this instrument, we offer the demonstration of visualizing the nanoscale periodic spatial modulation of an electron beam, which may be converted into longitudinal density modulation through emittance exchange to enable production of high-power coherent radiation at short wavelengths. Our results mark a great step towards single-shot nanometer-resolution MUEMs and compact intense x-ray sources that may have widespread applications in many areas of science.