Advances in Large Language Models (LLMs) have led to a surge of LLM-powered applications. These applications have diverse token-generation latency requirements. As a result, simply classifying workloads as latency-sensitive (LS) or best-effort (BE) overlooks the nuances within the latency-sensitive category and results in suboptimal user experiences and scheduling opportunities. However, efficiently serving requests with multiple SLO requirements poses significant challenges. First, all requests within a batch generate new tokens simultaneously, which can misalign them with their distinct SLO requirements. Moreover, while existing systems focus on auto-scaling for handling various overall request rates, the diversity of SLOs necessitates fine-grained auto-scaling among these SLO tiers. Finally, unlike LS/BE scenarios, where BE requests can be aborted at any time to ensure the SLO attainment of LS requests, those with different latency-sensitive SLOs cannot tolerate prolonged delays, and tail latency must be controlled. To tackle these challenges, we propose PolyServe, a novel multi-SLO scheduling policy at scale that maintains high SLO attainment while maximizing throughput. PolyServe first groups requests into multiple bins based on their per-token latency requirement, then schedules each bin to a subset of the server fleet. PolyServe routes requests to the highest-load but still SLO-attainable server to create a load gradient that facilitates auto-scaling. To increase utilization, PolyServe permits looser-SLO requests to share tighter-SLO instances when their own servers are saturated. PolyServe uses profiling data to guide scheduling decisions and manage tail latency through request-wait-time-aware scheduling, dynamic chunking, and continuous chunked prefill prediction. PolyServe achieves 1.23x goodput gain compared to existing policies, achieving up to 92.5% of optimal goodput.