What LIGO should detect that it has not yet? What would the lack of detection of such phenomena tell us about the Universe?
What should be the most frequent signals detected, theoretically speaking? I'm a lay person in the area, so you might have to give a very didactic answer...
 A: Good question. There are various categories of important findings.
One is related to the astronomy and astrophysics of black holes, and probably related phenomena. That includes how are those intermediate mass black holes formed (smaller ones of a few solar masses are understood, and supermassive ones partly, but not these 10-40 solar mass black holes), whether they are the source of gamma ray bursts and could that be black holes disappearing after Hawking radiation. Also finding any discrepancy with General Relativity such as whether black holes have any structure other than mass, change and rotation, or other possible violations of General Releativity (like dispersion of gravitational waves, which also was not seen in this last detection). Also neutron star structure and whether any are quark stars, supernova collapse dynamics, and others. Ths statistics are important not only to verify the observations, but more importantly to see if the numbers of black holes in the universe is what is expected, or if not if it indicates some cosmology factor we have not accounted for. See others at  https://www.ligo.caltech.edu/page/science-impact 
There is more also as we build others around the earth, and use them to correlate with LIGO to locate the exact sources, and thus marry the observations with radio and gamma ray observations, thus seeing the complete evolution. Also, as we build bigger ones, meaning interferometers in space with legs of 1000 Kms or more, we'll start seeing also the gravitational waves from supermassive black holes in the center of galaxies, or elsewhere, see their evolution and thus get some closure on galaxy formation. Even more, we might detect things that physics suspects might have occurred early in the universe (the first microseconds till about 380,000 years after the Big Bang. We could see the early effect of primordial Black Holes, and if they exist a first observation of cosmic strings and cosmic anomalies in the universe. This would have a large impact on particle physics, and for quantum gravity, indicating  what may be valid for energies and sizes way beyond what we can do in the LHC, or any super LHCs, or anything we can see in earth bound gravitational interferometers. We might also find something about the sources of dark energy and dark matter. 
Anything that proves more or disproves any part of General Relativity would be huge. Also, anything that gives us a hint of what very strong gravity and quantum gravity does would be a step to a still nonexistent a accepted theory of quantum gravity
See Scientific American more than a year old for some of these also, at https://www.scientificamerican.com/article/the-future-of-gravitational-wave-astronomy/. There should be more 'what's the future' articles coming out after this last detection. 
