Stars and planets are formed inside dense interstellar molecular clouds by processes imprinted on the three-dimensional (3D) morphology of the clouds. Determining the 3D structure of interstellar clouds remains challenging because of projection effects and difficulties measuring the extent of the clouds along the line of sight. We report the detection of normal vibrational modes in the isolated interstellar cloud Musca, allowing determination of the 3D physical dimensions of the cloud. We found that Musca is vibrating globally, with the characteristic modes of a sheet viewed edge on, not the characteristics of a filament as previously supposed. We reconstructed the physical properties of Musca through 3D magnetohydrodynamic simulations, reproducing the observed normal modes and confirming a sheetlike morphology.