Abstract: The Coons patch is a classical method for constructing smooth two-dimensional manifold mappings through boundary curve interpolation. It is widely applied in fields such as computer graphics, product shape design, and isogeometric analysis. However, due to factors such as complex boundary curve shapes and significant curvature variations, the Jacobian determinant of a Coons mapping may degenerate or even change sign in certain regions, causing a loss of regularity. This can severely affect the quality of the surface and compromise the numerical stability of subsequent analyses. Developing efficient and reliable methods for regularity verification is of great importance for achieving high-quality parameterizations. To address the regularity verification problem of spatial Coons patches, a series of theoretical conditions and algorithms is proposed. First, based on the condition of the rank of Jacobian matrix, a sufficient criterion for ensuring the regularity of Coons mappings is derived. Second, for Coons patches expressed in Bézier form, a regularity verification method based on the Bézier coefficients of the Jacobian determinant is proposed, transforming the verification task into a set of coefficient sign-consistency constraints, which significantly improves computational efficiency and geometric interpretability. Finally, a subdivision strategy for Bézier surfaces is introduced to construct a class of necessary and sufficient conditions, enabling the global verification problem to be reduced to local subdomain checks. The proposed Jacobian coefficient extraction framework can be extended to multi-patch B-spline surfaces with general topological structures, enhancing the generality and adaptability of the proposed method. Numerical experiment results show that for multi-patch B-spline surfaces consisting of 1 752 and 6 800 bi-cubic Bézier elements, the proposed algorithm completes the regularity verification within 0.112 seconds and 0.402 seconds, respectively, meeting the performance requirements of real-time engineering applications.