Rayleigh scattering spectra were obtained from individual single-walled carbon nanotubes with the use of a laser-generated visible and near-infrared supercontinuum. This diagnostic method is noninvasive and general for nanoscale objects. The approach permits clear identification of excited states in arbitrary metallic and semiconducting nanotubes. We analyzed spectral lineshapes in relation to the role of excitonic effects and correlated the results with Raman scattering data on individual tubes. The nanotube structure remained the same over distances of tens of micrometers. Small nanotube bundles retained distinct Rayleigh spectroscopic signatures of their component nanotubes, thus allowing the probing of nanotube-nanotube interactions.