The structural stability of the weakest bonded configurations of water clusters in the form of gas hydrate cavities D and T, as well as in the form of Kelvin’s polyhedron, was tested using the local geometric optimization with the polarizable force field AMOEBA. A number of different barrier-free molecular rearrangements were observed. Most of the configurations changed their shapes, forming surface defects of the same type. But the main attention is paid to the concerted molecular reorientations leading to the proton shift along hydrogen bonds. A number of statements about the topology of the hydrogen-bonded network are rigorously proved. It is concluded that the structural stability of configurations and the features of structural transformations are basically determined by the arrangement of the homodromic hydrogen-bonded water rings. The stability of the configurations, which retained their original shape during local geometric optimization, was studied by the molecular dynamics method with a gradual increase in temperature. Some interesting effects were found. This is the stabilizing locking of the most polarized configurations, as well as very intense amplitude oscillations of the free OH group at the end of two-bond flips.