Discrete spectrum of disturbances in high-speed boundary layers is discussed with emphasis on singularities caused by synchronization of the normal modes. Numerical examples illustrate different spectrum structures and jumps from one structure to another with small variations of basic-flow parameters. It is shown that this singular behavior is due to branching of the dispersion curves in the synchronization region. Depending on locations of the branch points, the spectrum contains one or two unstable modes. In this connection, the terminology used for instability of high-speed boundary layers is clarified. It is emphasized that the spectrum branching can cause significant difficulties in stability analyses based on traditional LST and PSE methods, which ignore the coupling between different modes. Multiple-mode considerations are needed to resolve these difficulties and obtain uniformly valid solutions.