Observations of X-ray binaries provide evidence for black holes formed from stellar collapse. An X-ray binary, so named because it emits X-ray radiation, consists of a dark compact object orbited by a normal star.
Pairs of stars often orbit each other in binary systems. If massive enough, one star of the pair may collapse, while the other member of the pair remains a normal star. A normal star orbiting closely around a black hole can transfer matter to the black hole. The temperature of the infalling matter increases as it approaches the black hole, until it glows strongly with X-rays that can often be detected easily by telescopes orbitng the Earth.
So, X-ray binaries supply prime candidates for black holes, but dark compact objects other than black holes might also emit X-rays. However, a compact object more massive than about two solar masses has too much mass to be either a white dwarf or neutron star, and consequently must be a black hole. Observations of the normal star's orbital velocity give estimates for the masses of both objects in an X-ray binary. Compact objects in several observed X-ray binaries have masses that fall in range required for black holes.
Observations of the orbital motion of stars and other objects about the central cores of galaxies suggest strongly that supermassive objects reside in galactic cores. The small size of galactic cores suggest strongly that the supermassive objects must be black holes. Our own galaxy, the Milky Way, has a three million solar mass black hole in its core.
No convincing observational evidence for primordial black holes presently exists, but hope exists. Hawking radiation causes black holes to evaporate. Primordial black holes that formed from small masses may, right now, be evaporating into a shower of high energy radiation. Scientists continue to hope that one day they will see such radiation.