There are roughly three methods to "discover" black holes:
- Find the x-ray signature of their accretion disk.
- Discover the gravitational wave signature of black hole mergers.
- Be clever and find something new, such as the radial velocity measurements that discovered the "unicorn".
The processing of gravitational waves is almost entirely automatic. The measurements of stress that are made by Ligo and Virgo are automatically processed to look for evidence of a "chirp", and candidates are automatically flagged for further investigation. GW discoveries currently don't exactly locate the black hole, but does discover their existence and give information on mass and distance.
X-ray sources are automatically located by X-ray nova searches, with space telescopes. The telescope scans a region of the sky, looking for changes, and flags possible sources for further investigation. The hard work is in showing that a particular X-ray source is a black hole and not a neutron star or something else. This tends to involve gathering a combination of evidence rather than a single "discovery".
Other methods, such as the radial velocity can also be processed automatically (by automatic systems looking for exoplanets). These can discover candidates, but gathering the evidence that the signal is due to a black hole is a human task.
Many "black holes" are in reality "black hole candidates". There is evidence of something, and that is consistent with a black hole, and so rather than a "discovery", there is a process whereby other hypotheses are eliminated by further observations, until only the black hole hypothesis remains.