The ICP registration algorithm has been a preferred method for LiDAR-based robot localization for nearly a decade. However, even in modern SLAM solutions, ICP can degrade and become unreliable in geometrically ill-conditioned environments. Current solutions primarily focus on utilizing additional sources of information, such as external odometry, to either replace the degenerate directions of the optimization solution or add additional constraints in a sensor-fusion setup afterward. In response, this work investigates and compares new and existing degeneracy mitigation methods for robust LiDAR-based localization and analyzes the efficacy of these approaches in degenerate environments for the first time in the literature at this scale. Specifically, this work proposes and investigates i) the incorporation of different types of constraints into the ICP algorithm, ii) the effect of using active or passive degeneracy mitigation techniques, and iii) the choice of utilizing global point cloud registration methods on the ill-conditioned ICP problem in LiDAR degenerate environments. The study results are validated through multiple real-world field and simulated experiments. The analysis shows that active optimization degeneracy mitigation is necessary and advantageous in the absence of reliable external estimate assistance for LiDAR-SLAM. Furthermore, introducing degeneracy-aware hard constraints in the optimization before or during the optimization is shown to perform better in the wild than by including the constraints after. Moreover, with heuristic fine-tuned parameters, soft constraints can provide equal or better results in complex ill-conditioned scenarios. The implementations used in the analysis of this work are made publicly available to the community.