In recent work, we presented mathematical theory and algorithms for time-frequency analysis of non-stationary signals. In that work, we generalized the definition of the Hilbert spectrum by using a superposition of complex AM-FM components parameterized by the Instantaneous Amplitude (IA) and Instantaneous Frequency (IF). Using our Hilbert Spectral Analysis (HSA) approach, the IA and IF estimates can be far more accurate at revealing underlying signal structure than prior approaches to time-frequency analysis. In this paper, we have applied HSA to speech and compared to both narrowband and wideband spectrograms. We demonstrate how the AM-FM components, assumed to be intrinsic mode functions, align well with the energy concentrations of the spectrograms and highlight fine structure present in the Hilbert spectrum. As an example, we show never before seen intra-glottal pulse phenomena that are not readily apparent in other analyses. Such fine-scale analyses may have application in speech-based medical diagnosis and automatic speech recognition (ASR) for pathological speakers.