Cognitive problems frequently accompany neurological manifestations of multiple sclerosis (MS). However, during screening of preclinical candidates, assessments of behaviour in mouse models of MS typically focus on locomotor activity. In the present study, we analysed cognitive behaviour of 9 to 10‐week‐old female C57Bl/6J mice orally administered with the toxin cuprizone that induces demyelination, a characteristic feature of MS. Animals received 400 mg/kg cuprizone daily for 2 or 4 weeks, and their performance was compared with that of vehicle‐treated mice. Cuprizone‐treated animals showed multiple deficits in short touchscreen‐based operant tasks: they responded more slowly to visual stimuli, rewards and made more errors in a simple rule‐learning task. In contextual/cued fear conditioning experiments, cuprizone‐treated mice showed significantly lower levels of contextual freezing than vehicle‐treated mice. Diffusion tensor imaging showed treatment‐dependent changes in fractional anisotropy as well as in axial and mean diffusivities in different white matter areas. Lower values of fractional anisotropy and axial diffusivity in cuprizone‐treated mice indicated developing demyelination and/or axonal damage. Several diffusion tensor imaging measurements correlated with learning parameters. Our results show that translational touchscreen operant tests and fear conditioning paradigms can reliably detect cognitive consequences of cuprizone treatment. The suggested experimental approach enables screening novel MS drug candidates in longitudinal experiments for their ability to improve pathological changes in brain structure and reverse cognitive deficits.