Context. Methanol has a rich rotational spectrum providing a large number of transitions at sub-millimetre wavelengths from a range of energy levels in one single telescope setting, thus making it a good tracer of physical conditions in star-forming regions. Furthermore, it is formed exclusively on grain surfaces and is therefore a clean tracer of surface chemistry. Aims. Determining the physical and chemical structure of low-mass, young stellar objects, in particular the abundance structure of CH₃OH, to investigate where and how CH₃OH forms and how it is eventually released back to the gas phase. Methods. Observations of the Serpens molecular core have been performed at the James Clerk Maxwell Telescope using the array receiver, Harp-B. Maps over a 4'5 × 5'4 region were made in a frequency window around 338 GHz, covering the 7_K-6_K transitions of methanol. Data are compared with physical models of each source based on existing sub-millimetre continuum data. Results. Methanol emission is extended over each source, following the column density of H₂ but showing up also particularly strongly around outflows. The rotational temperature is low, 15-20 K, and does not vary with position within each source. None of the Serpens Class 0 sources show the high-K lines seen in several other Class 0 sources. The abundance is typically 10^-9-10^-8 with respect to H₂ in the outer envelope, whereas "jumps" by factors of up to 10²-10³ inside the region where the dust temperature exceeds 100 K are not excluded. A factor of up to ~10³ enhancement is seen in outflow gas, consistent with previous studies. In one object, SMM4, the ice abundance has been measured to be ~ 3 × 10^-5 with respect to H₂ in the outer envelope, i.e., a factor of 10³ larger than the gas-phase abundance. Comparison with C¹⁸O J = 3-2 emission shows that strong CO depletion leads to a high gas-phase abundance of CH ₃OH not just for the Serpens sources, but also for a larger sample of deeply embedded protostars. Conclusions. The observations illustrate the large-scale, low-level desorption of CH₃OH from dust grains, extending out to and beyond 7500 AU from each source, a scenario which is consistent with non-thermal (photo-)desorption from the ice. The observations also illustrate the usefulness of CH₃OH as a tracer of energetic input in the form of outflows, where methanol is sputtered from the grain surfaces. Finally, the observations provide further evidence of CH₃OH formation through CO hydrogenation proceeding on grain surfaces in low-mass envelopes.