Hot subdwarfs (sdO/B) are the stripped helium cores of red giants formed via binary interactions. Close hot subdwarf binaries with massive white dwarf companions have been proposed as possible progenitors of thermonuclear supernovae type Ia (SN Ia). If the supernova is triggered by stable mass transfer from the helium star, the companion should survive the explosion and should be accelerated to high velocities. The hypervelocity star US 708 is regarded as the prototype for such an ejected companion. To find more of those objects we conducted an extensive spectroscopic survey. Candidates for such fast stars have been selected from the spectroscopic database of the Sloan Digital Sky Survey (SDSS) and several ground-based proper-motion surveys. Follow-up spectroscopy has been obtained with several 4m- to 10m-class telescopes. Combining the results from quantitative spectroscopic analyses with space-based astrometry from Gaia Early Data Release 3 (EDR3) we determined the atmospheric and kinematic parameters of 53 fast hot subdwarf stars. None of these stars is unbound to the Galaxy, although some have Galactic restframe velocities close to the Galactic escape velocity. 21 stars are apparently single objects, that crossed the Galactic disc within their lifetimes in the sdO/B stage and could be regarded as potential candidates for the SN Ia ejection scenario. However, the properties of the full sample are more consistent with a pure old Galactic halo population. We therefore conclude that the fast sdO/B stars we found are likely to be extreme halo stars.