Is that a neutron star or a black hole?

This question makes no sense if you are asking a normal astronomer since a black hole does not emit light while a neutron star does. However, this gets complicated when it comes to gravitational-wave observation.

When a pair of compact binaries revolve around each other, their orbital energy is slowly leaked to the rest of the universe through gravitational-wave. The signal mostly depends on the mass of the two compact objects in the binary system, and theoretically, the universe allows the existence of a "light" black hole which has the mass of a typical neutron star (< 3 solar mass). So if we have two binaries that share the same masses combination, then we cannot distinguish them from each other just by listening to the gravitational-wave they emit, at least not if we do not consider neutron stars are deformable while black holes are not.

Black holes are the stiffest object you can find in the universe, and they cannot be deformed at all. Neutron stars are also very stiff, but they can still be deformed if they are placed in a very strong gravitational tidal field, say, near another neutron star or a black hole. This softness, called tidal deformability, allows the neutron star to wobble when it goes around its companion, hence introducing a small modulation to the gravitational-wave signal. The larger the tidal deformability, the softer the neutron star, the stronger the modulation. If we can measure this modulation, doesn't this mean we can distinguish a neutron star from a black hole?

In principle yes, but with our current technology no. The current gravitational-wave detectors measure a combination of its components' deformability (effective tidal deformability), but not individual tidal deformability. Having only one variable measured, can we still determine whether a binary contains two neutron stars, or a neutron star and a black hole?

In "Distinguishing double neutron star from neutron star-black hole binary populations with gravitational-wave observations", we pointed out even though we cannot pin down whether an individual binary has a black hole or not, we can still figure this out probabilistically. Although we do not measure the tidal deformability of both components in a binary, we know the effective tidal deformability is likely to be higher if the system contains two neutron stars, because both of the components can contribute to the effective tidal deformability. With a population of binaries, we can compare the distribution of effective tidal deformability against our theoretical understanding, thus infer the mixing fraction between binary neutron stars and neutron star-black hole binary, i.e. the relative abundance of these two types of binary in our observed population. Given our model and our inference result on the mixing fraction, we can determine the probability of containing a black hole for a new binary (Figure 2 in the paper).

We predicted that we should have some clue about the mixing fraction with ~60 events laying in the mass range of interest. Of course, we took some simplistic assumptions such as we were marginalizing over the uncertainty in the equation of state (EOS), so the result we have is a bit optimistic. BUT! Taking these assumptions into account, we are still quite confident that we can start understanding the mixing fraction in a few years, given that we are expecting hundreds of events when the detectors in LSC operate at their design sensitivity soon.