We do not know for sure, 100%, that there are any black holes. However, beyond reasonable doubt, when the compact object in an X-ray binary system is shown to be more massive than about 3 times the mass of the Sun, then this compact object is a black hole.

With regard how many there are in total, it is thought that many, perhaps most; galaxies contain a massive black hole in their centers. If this is the case, then there will be almost as many black holes as galaxies - a great number!