Using molecular dynamics techniques, we examined equilibrium and dynamical characteristics pertaining to the solvation of a single beta-cyclodextrin (CD) in water and in dimethylsulfoxide (DMSO). Compared to its global minimum structure, the overall shape of the solute in solution is reasonably well preserved. While in aqueous solutions, the average number of solvent molecules retained within the central cavity of the oligosaccharide is close to 5, for DMSO, that number reduces to similar to 1. No evidence of significant orientational correlations of the trapped molecules were found in either solvent. The main contributions to the hydrogen-bond (HB) connectivity between the solute and the bulk phases are due to the more distal HO6-O6 hydroxyl groups, acting as HB donors and acceptors. The average residence time for retained DMSO was found to be in the nanosecond range, and it is, at least, 1 order of magnitude longer that the one observed for water. We also analyzed the characteristics of the solvation of the beta-CD in an equimolar water-DMSO mixture. In this environment, we found a preferential localization of a single DMSO molecule in the interior of the CD and a very minor retention of water. In the mixture, the characteristic time of residence of the trapped DMSO molecule increases by a factor of similar to 2. The observed difference was rationalized in terms of the fluctuations of the local concentrations of the two species in the vicinity of the CD top and bottom rims.