The recent, deepening penetration of variable energy resources (VERs) subjects power systems to the highly variable, rapidly changing and uncertain generation outputs that the operators must manage effectively to provide reliable service. Such variability poses additional challenges to the management of the continual variability in system loads. Consequently, the frequency regulation service requirements are substantially increased over those for systems without these VERs. The attractive rapid response and fast ramping capability characteristics of flywheel energy storage (FES) technology can be exploited in the FES deployment for frequency regulation service provision. Such deployment requires the effective management of the limited FES capacity and storage capability. In this paper, we construct optimized frequency regulation service offer strategies for an FES unit into the day-ahead markets (DAMs) and their associated real-time markets (RTMs). An independent grid operator (IGO) procures the service competitively through such offers. The FES offer strategies are obtained from the deployment of a robust optimization approach, which explicitly considers the uncertain nature of automatic generation control (AGC) signals and the amount of FES stored energy. The optimization objective is to maximize the FES service provision into the DAMs and the RTMs. To meet this objective, the FES offers only into particular DAMs and provides additional frequency regulation service in the RTMs by taking advantage of the most recent information of FES stored energy, even in those hours in which the FES does not participate in the DAMs. A salient characteristic of the proposed offer strategies is that all offered services are guaranteed to be provided, regardless of the actual AGC signals. We illustrate the effectiveness of the proposed strategies with simulation studies that use PJM Interconnection historical AGC signal data. Representative results indicate that the proposed FES offer strategies result in guaranteed service provision without any exceptions. Such service provision represents a major improvement over that under the conventional strategy that offers the full FES capacity into each DAM but satisfies, on average, only 47.7 % of the regulation service requests. We also study the impacts of the Federal Energy Regulatory Commission (FERC) Order No. 764 that allows DAM periods shorter than one hour. Specifically, we quantify the significant benefits emanating from the flexibility for limited storage capability devices such as FES units in frequency regulation service provision via simulation studies.