The interactions of methanol with well-defined Cu films on the oxygen-terminated ZnO(0001BAR)-O surface have been studied, mainly using temperature programmed desorption (TPD). The Cu films, which were from submonolayer to multilayer in coverage, had been structurally characterized in previous studies using XPS, LEIS, ARXPS, LEED and work function measurements, and by CO, H2O and formic acid adsorption. On clean Cu films methanol is adsorbed reversibly, desorbing at 200-260 K from atom-thick Cu islands, and at approximately 155 K from multilayer islands preannealed to approximately 550 K. In this respect, the atom-thin islands resemble Cu(110) sites and multilayer islands resemble Cu(111), consistent with behavior of other adsorbates. On oxygen-predosed multilayer films (preannealed to approximately 600 K), methanol reacts to form methoxy species which decompose at 395 K to yield formaldehyde and hydrogen in TPD, also like Cu(111). Multilayer films preannealed to > 750 K show a decrease in the peak area for methoxy decomposition which correlates with the loss of Cu area due to severe clustering. Oxygen-predosed Cu islands which are but one Cu atom thick show no clear evidence for a methoxy state in TPD. This suggests that oxygen atoms on such atom-thin Cu islands are poor Bronsted bases relative to O(a) on bulk Cu surfaces, consistent with results for adsorbed water. Results on high-area Cu/ZnO catalysts are discussed in the light of these new results.