Rapid advancements in foundation models, including Large Language Models, Vision-Language Models, Multimodal Large Language Models, and Vision-Language-Action Models, have opened new avenues for embodied AI in mobile service robotics. By combining foundation models with the principles of embodied AI, where intelligent systems perceive, reason, and act through physical interaction, mobile service robots can achieve more flexible understanding, adaptive behavior, and robust task execution in dynamic real-world environments. Despite this progress, embodied AI for mobile service robots continues to face fundamental challenges related to the translation of natural language instructions into executable robot actions, multimodal perception in human-centered environments, uncertainty estimation for safe decision-making, and computational constraints for real-time onboard deployment. In this paper, we present the first systematic review focused specifically on the integration of foundation models in mobile service robotics. We analyze how recent advances in foundation models address these core challenges through language-conditioned control, multimodal sensor fusion, uncertainty-aware reasoning, and efficient model scaling. We further examine real-world applications in domestic assistance, healthcare, and service automation, highlighting how foundation models enable context-aware, socially responsive, and generalizable robot behaviors. Beyond technical considerations, we discuss ethical, societal, and human-interaction implications associated with deploying foundation model-enabled service robots in human environments. Finally, we outline future research directions emphasizing reliability and lifelong adaptation, privacy-aware and resource-constrained deployment, and governance and human-in-the-loop frameworks required for safe, scalable, and trustworthy mobile service robotics.