  The study of deuteration in pre-stellar cores is important to understand the
physical and chemical initial 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 on surface processes at low temperatures. 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. Single-dish observations of CH$_3$OH, CH$_2$DOH, H$_2$CO,
H$_2\,^{13}$CO, HDCO, D$_2$CO and C$^{17}$O toward the prototypical pre-stellar
core L1544 were performed at the IRAM 30 m telescope. We analyze their column
densities, distributions, and compare these observations with gas-grain
chemical models. The maximum deuterium fraction derived for methanol is
[CH$_2$DOH]/[CH$_3$OH] $\sim$ 0.08$\pm$0.02, while the measured deuterium
fractions of formaldehyde at the dust peak are [HDCO]/[H$_2$CO] $\sim$
0.03$\pm$0.02, [D$_2$CO]/[H$_2$CO] $\sim$ 0.04$\pm$0.03 and [D$_2$CO]/[HDCO]
$\sim$ 1.2$\pm$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.
