3D point cloud data is increasingly used in safety-critical applications such as autonomous driving. Thus, robustness of 3D deep learning models against adversarial attacks is a major consideration. In this paper, we systematically study the impact of various self-supervised learning proxy tasks on different architectures and threat models for 3D point clouds. Specifically, we study MLP-based (PointNet), convolution-based (DGCNN), and transformer-based (PCT) 3D architectures. Through comprehensive experiments, we demonstrate that appropriate self-supervisions can significantly enhance the robustness in 3D point cloud recognition, achieving considerable improvements compared to the standard adversarial training baseline. Our analysis reveals that local feature learning is desirable for adversarial robustness since it limits the adversarial propagation between the point-level input perturbations and the model's final output. It also explains the success of DGCNN and the jigsaw proxy task in achieving 3D robustness.