The joint communications and sensing (JCAS) paradigm is envisioned as a core capability of sixth-generation (6G) wireless networks, enabling the integration of data communication and environmental sensing within a unified system. By reusing spectrum, waveforms, and hardware resources, JCAS improves spectral efficiency, reduces system complexity, and hardware cost, while enabling new use cases. Nevertheless, the realization of JCAS is hindered by inherent trade-offs between communication and sensing objectives, limited controllability of wireless propagation, and stringent hardware and design constraints. Simultaneously transmitting and reflecting reconfigurable intelligent surfaces (STAR-RIS) have recently emerged as a promising technology to address these challenges by enabling full-space programmable manipulation of electromagnetic waves. This survey provides a systematic and in-depth review of STAR-RIS-enabled JCAS systems. Specifically, we first introduce the fundamental principles of JCAS and STAR-RIS. We then classify and review the state-of-the-art research on STAR-RIS-assisted JCAS from multiple perspectives, encompassing system architectures, waveform and beamforming design, resource allocation, optimization frameworks, and learning-based control. Finally, we identify key open challenges that remain unsolved and outline promising future research directions toward intelligent, flexible, and perceptive 6G wireless networks.