Eigenloci crossing or veering in structural systems dependent on parameters, and having two nearly identical eigenvalues, is the subject of this paper. Utilizing some basic ideas from the perturbed bifurcation theory, a criterion for identifying eigenloci crossing or veering is presented along with a method for studying the modal properties of mistuned structural systems. Two classical continuous systems: a clamped rectangular membrane and a rotating guided circular string, have been chosen as illustrations. The analysis is performed by applying the singular perturbations technique for asymptotic expansions. These expansions are used to detect correctly, whether the eigenloci will cross over or veer away from a particular “singular” point under the influence of small parameter variations from nominal values. This is useful in the correct identification of the vibration behavior of mechanical systems subject to structural parameter variations, since it is known that only the eigenloci veering leads to modal bifurcations.