This paper presents the first high-resolution dynamic spectra of Saturn kilometric radiation acquired upon Cassini's approach and first orbits of Saturn. The emissions display upward and downward drifting features with bandwidths down to ~200 Hz and drift rates of a few kHz per second. At other times, the emissions are much more diffuse or continuous, showing little spectral structure on scales of 10 or 20 kHz. The fine structure is strikingly similar to Earth's auroral kilometric radiation (AKR) and Jovian auroral radio emissions in many respects. The dynamic spectral features provide insight into the highly nonlinear nature of the cyclotron maser instability believed to generate the emissions. We use ideas developed to explain the fine structures at Earth to suggest features and processes in the auroral acceleration region which may result in Saturn's fine structures.