This article proposes a robust and scalable clustering method for 3D point-cloud plane segmentation with applications in robotics, such as SLAM, collision avoidance, and object detection. Our approach builds on the sparse subspace clustering framework, which seeks a collection of subspaces that fit the data. Since subspace clustering requires solving a global sparse representation problem and forming a similarity graph, its high computational complexity is known to be a significant drawback, and performance is sensitive to a few hyperparameters. To tackle these challenges, our method has two key ingredients. We use randomized sampling to accelerate subspace clustering by solving a reduced optimization problem. We also analyze the obtained segmentation for quality assurance and performing a post-processing process to resolve two forms of model mismatch. We present numerical experiments to demonstrate the benefits and merits of our method.