I presume your course is asking you to explore fluid dynamics a bit. An inertial instability is turbulence due to exceeding certain velocities and dimensions (jointly) in your system. As a simplest example, laminar vs. turbulent flow in a pipe depending on the Reynolds Number. But a symmetric instability (though it's usually referred to as a conditional symmetric stability!) is turbulence or disturbance due to an interaction of a thermal gradient and a dynamic one, where each gradient would be stable if it were the only factor operating on the system (see the entry https://en.wikipedia.org/wiki/Conditional_symmetric_instability ).