A set of twelve different quasiperiodic Ammann grids with twelvefold-symmetric Fourier transform is constructed. Since the geometric dual of a dodecagonal Ammann grid is a quasiperiodic tiling which also has a twelvefold-symmetric Fourier transform, the twelve grids allow the construction of a set of twelve quasiperiodic tilings, all of which are presented here. Only one of them has been previously described. Nine of the tilings have a global center of twelvefold symmetry, while three have only bilateral global symmetry. The geometric properties of these twelve tilings such as shape and number of tiles, vertex configurations, and geometrical interrelations between different tilings from this set are described. The twelve tilings and their dual Ammann grids are found to belong to three classes of symmetry-preserving mutual local derivability (SMLD classes).