Most Type I superluminous supernovae (SLSNe-I) reported to date have been identified by their high peak luminosities and spectra lacking obvious signs of hydrogen. We demonstrate that these events can be distinguished from normal-luminosity SNe (including Type Ic events) solely from their spectra over a wide range of light-curve phases. We use this distinction to select 19 SLSNe-I and four possible SLSNe-I from the Palomar Transient Factory archive (including seven previously published objects). We present 127 new spectra of these objects and combine these with 39 previously published spectra, and we use these to discuss the average spectral properties of SLSNe-I at different spectral phases. We find that Mn II most probably contributes to the ultraviolet spectral features after maximum light, and we give a detailed study of the O II features that often characterize the early-time optical spectra of SLSNe-I. We discuss the velocity distribution of O II, finding that for some SLSNe-I this can be confined to a narrow range compared to relatively large systematic velocity shifts. Mg II and Fe II favor higher velocities than O II and C II, and we briefly discuss how this may constrain power-source models. We tentatively group objects by how well they match either SN 2011ke or PTF12dam and discuss the possibility that physically distinct events may have been previously grouped together under the SLSN-I label.