Mixture-of-Experts (MoE) has emerged as a powerful framework for multi-task learning (MTL). However, existing MoE-MTL methods often rely on single-task pretrained backbones and suffer from redundant adaptation and inefficient knowledge sharing during the transition from single-task to multi-task learning (STL to MTL). To address these limitations, we propose adaptive shared experts (ASE) within a low-rank adaptation (LoRA) based MoE, where shared experts are assigned router-computed gating weights jointly normalized with sparse experts. This design facilitates STL to MTL transition, enhances expert specialization, and cooperation. Furthermore, we incorporate fine-grained experts by increasing the number of LoRA experts while proportionally reducing their rank, enabling more effective knowledge sharing under a comparable parameter budget. Extensive experiments on the PASCAL-Context benchmark, under unified training settings, demonstrate that ASE consistently improves performance across diverse configurations and validates the effectiveness of fine-grained designs for MTL.