Context. The study of deuteration in pre-stellar cores is important in order to understand the initial physical and chemical conditions in the process of star formation. In particular, observations toward pre-stellar cores of methanol and deuterated methanol, solely formed on the surface of dust grains, may provide useful insights into surface processes at low temperatures. Aims. Here we analyze maps of CO, methanol, formaldehyde, and their deuterated isotopologues toward a well-known pre-stellar core. This study allows us to test current gas–dust chemical models. Methods. Single-dish observations of CH₃OH, CH₂DOH, H₂CO, H₂¹³ CO, HDCO, D₂CO, and C¹⁷O toward the prototypical pre-stellar core L1544 were performed at the IRAM 30 m telescope. We analyze their column densities and distributions, and compare these observations with gas–grain chemical models. Results. The maximum deuterium fraction derived for methanol is [CH₂DOH]/[CH₃OH] ~ 0.08 ± 0.02, while the measured deuterium fractions of formaldehyde at the dust peak are [HDCO]/[H₂CO] ~ 0.03 ± 0.02, [D₂CO]/[H₂CO] ~ 0.04 ± 0.03, and [D₂CO]/[HDCO] ~ 1.2 ± 0.3. Observations differ significantly from the predictions of models, finding discrepancies between a factor of 10 and a factor of 100 in most cases. It is clear though that to efficiently produce methanol on the surface of dust grains, quantum tunneling diffusion of H atoms must be switched on. It also appears that the currently adopted reactive desorption efficiency of methanol is overestimated and/or that abstraction reactions play an important role. More laboratory work is needed to shed light on the chemistry of methanol, an important precursor of complex organic molecules in space.