In this work, we report multispectral superconducting nanowire single photon detectors (SNSPDs) that can simultaneously detect single photons at multiple wavelengths with high efficiency. The superconducting nanowires are fabricated on an all-dielectric mirror consisting of two quarter-wave stack reflectors with separated central wavelengths. The unique optical structure results in serially coupled optical cavities, leading to multiple resonant absorption bands that are utilized for high-efficiency single photon detection. The fabricated detector shows system detection efficiencies of >80% at the three target wavelengths of 1550 nm, 1310 nm, and 1064 nm. The multispectral detector may eliminate the need for multiple SNSPDs for different wavelengths in a system, potentially resulting in a reduction in size, weight, and power, as well as in the cost of the overall detection system. The detector may also find interesting use for applications such as multispectral ranging or imaging.