Planes are fundamental primitives of 3D sences, especially in man-made environments such as indoor spaces and urban streets. Representing these planes in a structured and parameterized format facilitates scene editing and physical simulations in downstream applications. Recently, Gaussian Splatting (GS) has demonstrated remarkable effectiveness in the Novel View Synthesis task, with extensions showing great potential in accurate surface reconstruction. However, even state-of-the-art GS representations often struggle to reconstruct planar regions with sufficient smoothness and precision. To address this issue, we propose GSPlane, which recovers accurate geometry and produces clean and well-structured mesh connectivity for plane regions in the reconstructed scene. By leveraging off-the-shelf segmentation and normal prediction models, GSPlane extracts robust planar priors to establish structured representations for planar Gaussian coordinates, which help guide the training process by enforcing geometric consistency. To further enhance training robustness, a Dynamic Gaussian Re-classifier is introduced to adaptively reclassify planar Gaussians with persistently high gradients as non-planar, ensuring more reliable optimization. Furthermore, we utilize the optimized planar priors to refine the mesh layouts, significantly improving topological structure while reducing the number of vertices and faces. We also explore applications of the structured planar representation, which enable decoupling and flexible manipulation of objects on supportive planes. Extensive experiments demonstrate that, with no sacrifice in rendering quality, the introduction of planar priors significantly improves the geometric accuracy of the extracted meshes across various baselines.