The Iterative Closest Point (ICP) algorithm is a crucial component of LiDAR-based SLAM algorithms. However, its performance can be negatively affected in unstructured environments that lack features and geometric structures, leading to low accuracy and poor robustness in localization and mapping. It is known that degeneracy caused by the lack of geometric constraints can lead to errors in 6-DOF pose estimation along ill-conditioned directions. Therefore, there is a need for a broader and more fine-grained degeneracy detection and handling method. This paper proposes a new point cloud registration framework, LP-ICP, that combines point-to-line and point-to-plane distance metrics in the ICP algorithm, with localizability detection and handling. Rather than relying solely on point-to-plane localizability information, LP-ICP enhances the localizability analysis by incorporating a point-to-line metric, thereby exploiting richer geometric constraints. It consists of a localizability detection module and an optimization module. The localizability detection module performs localizability analysis by utilizing the correspondences between edge points (with low local smoothness) to lines and planar points (with high local smoothness) to planes between the scan and the map. The localizability contribution of individual correspondence constraints can be applied to a broader range. The optimization module adds additional soft and hard constraints to the optimization equations based on the localizability category. This allows the pose to be constrained along ill-conditioned directions. The proposed method is evaluated on simulation and real-world datasets, showing comparable or better accuracy than the state-of-the art methods in tested scenarios. Observed variations in partially localizable directions suggest the need for further investigation on robustness and generalizability.