The existing methods for reconfigurable intelligent surface (RIS) beamforming in wireless communications are typically limited to uniform phase quantization. However, in practical applications, engineering challenges and design requirements often lead to non-uniform phase and bit resolution of RIS units, which limits the performance potential of these methods. To address this issue, this paper pioneers the study of discrete non-uniform phase configuration in RIS-assisted multiple-input single-output (MISO) communication and formulates an optimization model to characterize the problem. For single-user scenarios, the paper proposes a partition-andtraversal (PAT) algorithm that efficiently achieves the global optimal solution through systematic search and traversal. For larger-scale multi-user scenarios, aiming to balance performance and computational complexity, an enhanced PAT-based algorithm (E-PAT) is developed. By optimizing the search strategy, the E-PAT algorithm significantly reduces computational overhead and achieves linear complexity. Numerical simulations confirm the effectiveness and superiority of the proposed PAT and EPAT algorithms. Additionally, we provide a detailed analysis of the impact of non-uniform phase quantization on system performance.