3D Semantic Scene Completion (SSC) has gained increasing attention due to its pivotal role in 3D perception. Recent advancements have primarily focused on refining voxel-level features to construct 3D scenes. However, treating voxels as the basic interaction units inherently limits the utilization of class-level information, which is proven critical for enhancing the granularity of completion results. To address this, we propose Disentangling Instance and Scene Contexts (DISC), a novel dual-stream paradigm that enhances learning for both instance and scene categories through separated optimization. Specifically, we replace voxel queries with discriminative class queries, which incorporate class-specific geometric and semantic priors. Additionally, we exploit the intrinsic properties of classes to design specialized decoding modules, facilitating targeted interactions and efficient class-level information flow. Experimental results demonstrate that DISC achieves state-of-the-art (SOTA) performance on both SemanticKITTI and SSCBench-KITTI-360 benchmarks, with mIoU scores of 17.35 and 20.55, respectively. Remarkably, DISC even outperforms multi-frame SOTA methods using only single-frame input and significantly improves instance category performance, surpassing both single-frame and multi-frame SOTA instance mIoU by 17.9% and 11.9%, respectively, on the SemanticKITTI hidden test.