One of the most intriguing ongoing searches (at least for me) is the gravitational wave (GW) detection efforts being done by LIGO collaboration. We already know about at least two black hole merger events. These guys are really interesting, since they serve as a good general relativity test. There are a bunch of other things that we can constraint with the statistics of these events... However, there are even more exciting things, that we can find with a GW telescope.
We still haven't seen any neutron-star with black hole or neutron star with neutron star mergers. That's a little weird, since there should be tons of them (according to predictions). It seems like these events are below the sensitivity limit.
Why these things will be exciting to catch? Because unlike black holes, neutron stars are made of matter, and this matter is in the most extreme conditions in our Universe, with overwhelmingly large magnetic fields, pressures and densities deep inside the core. The problem is that we still don't know what the neutron star interior looks like, what is it made of. It would be really nice to look deep inside, to somehow break the neutron star apart... And that's when merging processes come in.
When two neutron stars collide or when a neutron star falls into a black hole, during the very last seconds of this collision the neutron star squeezes and breaks apart, extracting an enormous amount of energy in the form of an electromagnetic radiation. If we're able to recognize this electromagnetic radiation (caught with some gamma-ray or x-ray telescope) just before the merging event, which will be signified by the GW telescope, we'll have a really good chance to find out what a neutron star is made of.
UPD (10 Nov, 2017). We've already seen neutron star with neutron star merger. :)