Reliable prediction is an essential requirement for deep neural models that are deployed in open environments, where both covariate and semantic out-of-distribution (OOD) data arise naturally. In practice, to make safe decisions, a reliable model should accept correctly recognized inputs while rejecting both those misclassified covariate-shifted and semantic-shifted examples. Besides, considering the potential existing trade-off between rejecting different failure cases, more convenient, controllable, and flexible failure detection approaches are needed. To meet the above requirements, we propose a simple failure detection framework to unify and facilitate classification with rejection under both covariate and semantic shifts. Our key insight is that by separating and consolidating failure-specific reliability knowledge with low-rank adapters and then integrating them, we can enhance the failure detection ability effectively and flexibly. Extensive experiments demonstrate the superiority of our framework.