The "instantaneous" normal mode (INM) spectrum for a model soft-sphere system is calculated via molecular-dynamics computer simulation for a variety of temperatures both above and below the observed glass transition temperature. In such spectra, the modes can be classified as either real or imaginary frequency. Results indicate that the liquid \ensuremath{\rightarrow} glass transition is associated with the temperature below which all "unstable" INM's become localized and that the crossover from unstable to stable INM's occurs at a finite imaginary frequency and not at zero as is generally assumed. Two possible mechanisms for the glass transition are discussed that are consistent with the present data.