Research trends in SLAM systems are now focusing more on multi-sensor fusion to handle challenging and degenerative environments. However, most existing multi-sensor fusion SLAM methods mainly use all of the data from a range of sensors, a strategy we refer to as the all-in method. This method, while merging the benefits of different sensors, also brings in their weaknesses, lowering the robustness and accuracy and leading to high computational demands. To address this, we propose a new fusion approach -- Selective Kalman Filter -- to carefully choose and fuse information from multiple sensors (using LiDAR and visual observations as examples in this paper). For deciding when to fuse data, we implement degeneracy detection in LiDAR SLAM, incorporating visual measurements only when LiDAR SLAM exhibits degeneracy. Regarding degeneracy detection, we propose an elegant yet straightforward approach to determine the degeneracy of LiDAR SLAM and to identify the specific degenerative direction. This method fully considers the coupled relationship between rotational and translational constraints. In terms of how to fuse data, we use visual measurements only to update the specific degenerative states. As a result, our proposed method improves upon the all-in method by greatly enhancing real-time performance due to less processing visual data, and it introduces fewer errors from visual measurements. Experiments demonstrate that our method for degeneracy detection and fusion, in addressing degeneracy issues, exhibits higher precision and robustness compared to other state-of-the-art methods, and offers enhanced real-time performance relative to the all-in method. The code is openly available.