Underwater sensor networks (UWSNs) have been investigated in a variety of applications such as sea resources reconnaissance, pollution monitoring and tactical monitoring. In 3D underwater environments, it is a key topic to judge the localizability of sensor nodes given known locations of a small set of anchor nodes. In this paper, a novel localizability judgment method for UWSNs is proposed based on rigidity theory. A UWSN is modelled as an undirected graph based on acoustic connectivity. The graph is then reduced to a subgraph with global rigidity, called skeleton, from which the set of localizable sensors can determined. Furthermore, the Analytic Hierarchy Process (AHP) is used to evaluate the localization confidence of localizable sensors. Extensive simulations demonstrate that the proposed localizability judgment method can achieve low false negative rate and high efficiency networks of different sensor numbers and sensor densities. It is also shown to perform well in dynamic networks with relatively low waterflow speed.