Slug tests are one of the most common field methods for estimating local hydraulic properties, for fast and low-cost characterization of aquifer heterogeneity. In highly permeable fracture zones, underdamped responses, identified by oscillations of the water level, are generally observed. Several analytical and numerical solutions have been developed for modeling underdamped slug test responses. However, difficulties of interpreting data in fractured rocks from such tests are generally encountered because the complexity of the system raises the question of the choice of the appropriate model. In order to gain more insights about the fundamental problem of uniqueness arising with slug test analyses in heterogeneous media, sensitivity and uniqueness analysis of three transient-flow models of underdamped slug tests is proposed. As different flow geometries may be encountered in fractured rocks, we propose to evaluate solutions for linear, radial and spherical flows including inertial and wellbore skin effects in a fully penetrating well. As expected, the results from sensitivity analysis show that the hydraulic conductivity parameter is the most sensitive parameter regardless of model configurations. However, the sensitivity to aquifer storage is important for the linear flow case, less important for the radial case and negligible for the spherical case. The wellbore skin is also not negligible and must be considered. The uniqueness analyses show that underdamped responses are challenging to interpret as different flow geometry may fit the data, which results in different estimation of the permeability.