This paper addresses the problem of accurate localization for quadrupedal robots operating in narrow tunnel-like environments. Due to the long and homogeneous characteristics of such scenarios, LiDAR measurements often provide weak geometric constraints, making traditional sensor fusion methods susceptible to accumulated motion estimation errors. To address these challenges, we propose AIMS, an adaptive LiDAR-IMU-leg odometry fusion method for robust quadrupedal robot localization in degenerate environments. The proposed method is formulated within an error-state Kalman filtering framework, where LiDAR and leg odometry measurements are integrated with IMU-based state prediction, and measurement noise covariance matrices are adaptively adjusted based on online degeneracy-aware reliability assessment. Experimental results obtained in narrow corridor environments demonstrate that the proposed method improves localization accuracy and robustness compared with state-of-the-art approaches.