We present redshifts for 2753 low-redshift (0.05~<z_spec_~<0.5) galaxies with 18<=r<=22 obtained with Hectospec at the Multi-Mirror Telescope (MMT). The observations targeted the XMM-LSS, ELAIS-N1 and DEEP2-3 fields, each of which covers ~1deg^2^. These fields are also part of the recently completed CFHT Large Area U-band Deep Survey (CLAUDS) and on-going Hyper Suprime-Cam deep fields surveys. The efficiency of our technique for selecting low-redshift galaxies is confirmed by the redshift distribution of our sources. In addition to redshifts, these high S/N spectra are used to measure ages, metallicities, and nuclear activity levels. In combination with the photometric catalogue in u, g, r, i, z, y down to 27 AB mag, we are able to study the galaxy population down to stellar masses of ~10^8^M_{sun}_ . This paper presents the observation strategy, the reduction procedure and properties of the galaxy sample. The catalog can be accessed through the webpage of this survey http://mips.as.arizona.edu/~cnaw/Faint_Low_z/.