Noise injection-based regularization, such as Dropout, has been widely used in image domain to improve the performance of deep neural networks (DNNs). However, efficient regularization in the point cloud domain is rarely exploited, and most of the state-of-the-art works focus on data augmentation-based regularization. In this paper, we, for the first time, perform systematic investigation on noise injection-based regularization for point cloud-domain DNNs. To be specific, we propose a series of regularization techniques, namely DropFeat, DropPoint and DropCluster, to perform noise injection on the point feature maps at the feature level, point level and cluster level, respectively. We also empirically analyze the impacts of different factors, including dropping rate, cluster size and dropping position, to obtain useful insights and general deployment guidelines, which can facilitate the adoption of our approaches across different datasets and DNN architectures. We evaluate our proposed approaches on various DNN models for different point cloud processing tasks. Experimental results show our approaches enable significant performance improvement. Notably, our DropCluster brings 1.5%, 1.3% and 0.8% higher overall accuracy for PointNet, PointNet++ and DGCNN, respectively, on ModelNet40 shape classification dataset. On ShapeNet part segmentation dataset, DropCluster brings 0.5%, 0.5% and 0.2% mean Intersection-over-union (IoU) increase for PointNet, PointNet++ and DGCNN, respectively. On S3DIS semantic segmentation dataset, DropCluster improves the mean IoU of PointNet, PointNet++ and DGCNN by 3.2%, 2.9% and 3.7%, respectively. Meanwhile, DropCluster also enables the overall accuracy increase for these three popular backbone DNNs by 2.4%, 2.2% and 1.8%, respectively.