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Using the largest spectroscopic data set of stripped-envelope core-collapse supernovae (stripped SNe), we present a systematic investigation of spectral properties of Type IIb SNe (SNe IIb), Type Ib SNe (SNe Ib), and Type Ic SNe (SNe Ic). Prior studies have been based on individual objects or small samples. Here, we analyze 242 spectra of 14 SNe IIb, 262 spectra of 21 SNe Ib, and 207 spectra of 17 SNe Ic based on the stripped SN data set of Modjaz et al. and other published spectra of individual SNe. Each SN in our sample has a secure spectroscopic ID, a date of V-band maximum light, and most have multiple spectra at different phases. We analyze these spectra as a function of subtype and phase in order to improve the SN identification scheme and constrain the progenitors of different kinds of stripped SNe. By comparing spectra of SNe IIb with those of SNe Ib, we find that the strength of Hα can be used to quantitatively differentiate between these two subtypes at all epochs. Moreover, we find a continuum in observational properties between SNe IIb and Ib. We address the question of hidden He in SNe Ic by comparing our observations with predictions from various models that either include hidden He or in which He has been burnt. Our results favor the He-free progenitor models for SNe Ic. Finally, we construct continuum-divided average spectra as a function of subtype and phase to quantify the spectral diversity of the different types of stripped SNe.