This is the first in a series of gamma-ray burst spectroscopy catalogs from the Burst and Transient Source Experiment (BATSE) on the Compton Gamma Ray Observatory, each covering a different aspect of burst phenomenology. In this paper, we present time sequences of spectral fit parameters for 156 bursts selected for either their high peak flux or fluence. All bursts have at least eight spectra in excess of 45{sigma} above background and span burst durations from 1.66 to 278s. Individual spectral accumulations are typically 128ms long at the peak of the brightest events but can be as short as 16ms, depending on the type of data selected. We have used mostly high energy resolution data from the Large Area Detectors, covering an energy range of typically 28-1800keV. The spectral model chosen is from a small empirically determined set of functions, such as the well-known "GRB" function, that best fits the time-averaged burst spectra. Thus, there are generally three spectral shape parameters available for each of the 5500 total spectra: a low-energy power-law index, a characteristic break energy, and possibly a high-energy power-law index. We present the distributions of the observed sets of these parameters and comment on their implications.