We report a novel experimental method to measure nanometer displacements using wavelength shifts of spectral peaks around spectral switch or singular phase points in the interference spectra due to temporal correlation in a Michelson interferometer illuminated by a broadband white-light source. Dramatic changes in the spectral characteristics are recorded as a function of path difference between the interfering beams around the spectral switch position. These are then compared with measurements far from it in order to demonstrate the higher sensitivities involved in the proposed method.