Recent observations indicate that organic molecules are prevalent towards starless and prestellar cores. Deuteration of these molecules has not been well-studied during the starless phase. Published observations of singly-deuterated methanol, CH2DOH, have only been observed in a couple of well-studied, dense and evolved prestellar cores (e.g. L1544, L183). Since the formation of gas-phase methanol during this cold phase is believed to occur via desorption from the icy grain surfaces, observations of CH2DOH may be useful as a probe of the deuterium fraction in the ice mantles of dust grains. We present a systematic survey of CH2DOH towards 12 starless and prestellar cores in the B10 region of the Taurus Molecular Cloud. Nine of the twelve cores are detected with [CH2DOH]/[CH3OH] ranging from <0.04 to 0.23$^{+0.12}_{-0.06}$ with a median value of 0.11. Sources not detected tend to have larger virial parameters and larger methanol linewidths than detected sources. The results of this survey indicate that deuterium fractionation of organic molecules, such as methanol, during the starless phase may be more easily detectable than previously thought.