'High-velocity features' (HVFs) are spectral features in Type Ia supernovae (SNe Ia) that have minima indicating significantly higher (by greater than about 6000km/s) velocities than typical 'photospheric-velocity features' (PVFs). The PVFs are absorption features with minima indicating typical photospheric (i.e. bulk ejecta) velocities (usually~9000-15000km/s near B-band maximum brightness). In this work, we undertake the most in-depth study of HVFs ever performed. The data set used herein consists of 445 low-resolution optical and near-infrared (NIR) spectra (at epochs up to 5d past maximum brightness) of 210 low-redshift SNe Ia that follow the 'Phillips relation'. A series of Gaussian functions is fit to the data in order to characterize possible HVFs of CaII H&K, SiII {lambda}6355, and the CaII NIR triplet. The temporal evolution of the velocities and strengths of the PVFs and HVFs of these three spectral features is investigated, as are possible correlations with other SN Ia observables. We find that while HVFs of CaII are regularly observed (except in underluminous SNe Ia, where they are never found), HVFs of Siii {lambda}6355 are significantly rarer, and they tend to exist at the earliest epochs and mostly in objects with large photospheric velocities. It is also shown that stronger HVFs of SiII {lambda}6355 are found in objects that lack CII absorption at early times and that have red ultraviolet/optical colours near maximum brightness. These results lead to a self-consistent connection between the presence and strength of HVFs of SiII {lambda}6355 and many other mutually correlated SN Ia observables, including photospheric velocity.