The use of digital watermarking in real applications is impeded by the weakness of current available algorithms against signal processing manipulations leading to the desynchronization of the watermark embedder and detector. For this reason, the problem of watermarking under geometric attacks has received considerable attention throughout recent years. Despite their importance, only few classes of geometric attacks are considered in the literature, most of which consist of global geometric attacks. The random bending attack contained in the Stirmark benchmark software is the most popular example of a local geometric transformation. In this paper, we introduce two new classes of local desynchronization attacks (DAs). The effectiveness of the new classes of DAs is evaluated from different perspectives including perceptual intrusiveness and desynchronization efficacy. This can be seen as an initial effort towards the characterization of the whole class of perceptually admissible DAs, a necessary step for the theoretical analysis of the ultimate performance reachable in the presence of watermark desynchronization and for the development of a new class of watermarking algorithms that can efficiently cope with them.