It has been found that the geometrical diagnostic methods can break the degeneracy for dark energy models. In this paper, we investigate the $Om$ diagnostic, the statefinder hierarchy $S_{n}$ and the composite null diagnostic ${S_{n},epsilon}$ for the Tsallis holographic dark energy models with interactions. We find that model parameters and the forms of interaction will influence the values of diagnostic parameters or the trends of the evolutionary trajectories for each model. Moreover, the statefinder hierarchy $S_{3}^{(1)}$ together with ${S_{3}^{(1)},epsilon}$ could give good diagnostic results. Furthermore, we also obtain some issues of cosmological structure by means of the composite null diagnostic.
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