Unmanned aerial vehicle (UAV)-based integrated sensing and communication (ISAC) systems are poised to revolutionize next-generation wireless networks by enabling simultaneous sensing and communication (S\&C). This survey comprehensively reviews UAV-ISAC systems, highlighting foundational concepts, key advancements, and future research directions. We explore recent advancements in UAV-based ISAC systems from various perspectives and objectives, including advanced channel estimation (CE), beam tracking, and system throughput optimization under joint sensing and communication S\&C constraints. Additionally, we examine weighted sum rate (WSR) and sensing trade-offs, delay and age of information (AoI) minimization, energy efficiency (EE), and security enhancement. These applications highlight the potential of UAV-based ISAC systems to improve spectrum utilization, enhance communication reliability, reduce latency, and optimize energy consumption across diverse domains, including smart cities, disaster relief, and defense operations. The survey also features summary tables for comparative analysis of existing methodologies, emphasizing performance, limitations, and effectiveness in addressing various challenges. By synthesizing recent advancements and identifying open research challenges, this survey aims to be a valuable resource for developing efficient, adaptive, and secure UAV-based ISAC systems.