The structure of indium-catalyzed germanium nanowires is investigated by atomic force microscopy, scanning confocal Raman spectroscopy and transmission electron microscopy. The nanowires are formed by a crystalline core and an amorphous shell. We find that the diameter of the crystalline core varies along the nanowire, down to few nanometers. Phonon confinement effects are observed in the regions where the crystalline region is the thinnest. The results are consistent with the thermally insulating behavior of the core-shell nanowires.