The paper deals with stability problems of linear gyroscopic systems with finite or infinite degrees of freedom, where the system matrices or operators depend smoothly on several real parameters. Explicit formulae for the behavior of eigenvalues under a change of parameters are obtained. It is shown that the bifurcation (splitting) of double eigenvalues is closely related to the stability, flutter and divergence boundaries in the parameter space. Normal vectors to these boundaries are derived using only information at a boundary point: eigenvalues, eigenvectors and generalized eigenvectors, as well as first derivatives of the system matrices (or operators) with respect to parameters. These results provide simple and constructive stability and instability criteria. The presented theory is exemplified by two mechanical problems: a rotating elastic shaft carrying a disk, and an axially moving tensioned beam.