Hydro-viscous drive (HVD) plays a significant role in smoothly transferring torque and flexibly regulating the velocity of the disks. By hydro-viscous drive, we mean that the viscous shear stress of the thin oil film between a multi-layer assembly of rotating parallel disks is generated to transmit torque and power. The laminar-to-turbulent transition is an extremely complicated issue due to the combined effects of squeeze and shear on the oil film within the microscale friction pair system. Hence, a comprehensive and thorough analysis of flow instability in fluid-thermal-solid interaction of tribodynamic behavior is highly desirable. Following a brief introduction of fundamentals of HVD, this paper provides an overall review on the instability mechanisms for three types of canonical flow dynamic models, i.e., plane squeeze flow, plane shear flow, and rotating-disk flow. The effects of various aspects of wall conditions and working media, such as surface microstructure, and temperature-dependent viscosity, on flow instability are then summarized, which can serve as a reference and guidance for optimizing the design of friction pair systems. Based on the review of the former progress, this paper not only explores the in-depth mechanisms regarding the laminar-to-turbulent transition in microchannel flow, but also provides the possibility of bridging the gap between flow instability and tribodynamic behavior.