In this paper we explore experimentally the folding and unfolding of creases of thin sheets of polymer materials and corresponding weaved fabric with a multiscale structure. A method to form reproducible folds is first proposed, which allows extraction of the fold’s properties in particular its maximum curvature versus the material and sheet properties, in full agreement with an elasto-plastic model of the folding process. Then we explore the free unfolding of a fold and demonstrate a time-dependent logarithmic reopening, interpreted in terms of a plastic creep of the material under the stored elastic stress. Finally we consider the effect of humidity on this unfolding process. We demonstrate that in strong contrast to a bulk sheet the weaved fabric made of the same material exhibits a strong humidity dependence, with humidity speeding up the logarithmic unfolding process. This phenomenon, which exhibits a close analogy to the previously observed humidity induced ageing of granular material properties, points to the observed key role of hygrometry in the unfolding of multiscale materials, as observed during ironing.