The Moore-Greitzer partial differential equation (PDE) is a commonly used parameter-dependent mathematical model for capturing flow disturbances in axial-flow jet engine compressors. For a certain compressor configuration, as the throttle coefficient (the parameter) decreases, inlet flow disturbances dominate, and stall occurs. Instabilities place fundamental limits on jet engine operating range and thus limit the design space. In the presence of stochastic multiplicative noise, in contrast to the original deterministic PDE model, almost-sure asymptotic stability is not well understood. To control and mitigate dynamic instabilities in modern jet engine compressors, the detection of the critical point where the system loses stability becomes essential. In this paper, we take advantage of recent advances in utilizing multiscale analysis to approximate the maximal Lyapunov exponent, which can readily serve as an indicator of stability change, and apply the results to study the impact of small multiplicative noise on compression system instabilities. A comprehensive review of related works is also provided.