Establishing character shape correspondence is a critical and fundamental task in computer vision and graphics, with diverse applications including re-topology, attribute transfer, and shape interpolation. Current dominant functional map methods, while effective in controlled scenarios, struggle in real situations with more complex challenges such as non-isometric shape discrepancies. In response, we revisit registration-for-correspondence methods and tap their potential for more stable shape correspondence estimation. To overcome their common issues including unstable deformations and the necessity for pre-alignment or high-quality initial 3D correspondences, we introduce Stable-SCore: A Stable Registration-based Framework for 3D Shape Correspondence. We first re-purpose a foundation model for 2D character correspondence that ensures reliable and stable 2D mappings. Crucially, we propose a novel Semantic Flow Guided Registration approach that leverages 2D correspondence to guide mesh deformations. Our framework significantly surpasses existing methods in more challenging scenarios, and brings possibilities for a wide array of real applications, as demonstrated in our results.