The visible binary member would appear to wobble. Astronomers are interested in stars that move for their own reasons, so they would track this wobble. It would be noticed that the star is following an orbit around another star, just like a binary system. The only problem is that there is no visible partner. That in itself is not enough to say that the partner is a black hole.
Many binary systems involve neutron stars.
Well, there are three scenarios. When binary stars orbit one another, sometimes they get close enough that they steal each other's atmosphere. It happens readily enough. Eventually, one of the stars supernovas, leaving a white dwarf, a neutron star, or a black hole. That's the first scenario.