* How do SMBHs form? Is it from the merger of a bunch of stellar-mass black holes or do is it from the collapse of a huge gas cloud 1000s of times the mass of the Sun (or more) when the galaxy is first forming?
* Why are the masses of SMBHs so tightly correlated to the masses of their host galaxies? This is known as the M-sigma relation and there's been a lot of research on this topic, in particular. The existence of this correlation implies that SMBHs are somehow involved in some sort of feedback mechanism with the rest of the galaxy, but this is very poorly understood. (There have been lots of different feedback mechanisms invoked, but it's been hard to discriminate between them to see what's really going on.)
* How do SMBHs merge? We know that SMBHs are at the center of nearly all galaxies (if not all of them), and we know that galaxies merge. We don't usually see multiple SMBHs in a galaxy, so somehow the SMBHs must merge as well. But models of these mergers stall out at around a few light-years. At this distance, stars are too far away to influence the orbit, but gravitational waves are too weak to do anything. Somehow real SMBH binaries in the universe manage to overcome this problem, but how? This is known as the last parsec problem.
* We see SMBHs driving huge, extremely powerful jets. What causes the formation of these jets? Why are they so collimated? The Blandford-Znajek process is usually invoked, but (at least in my opinion) while it's plausible, there's a fair amount of hand-waving and not as much observational evidence as one would like.
* Not a question dealing with SMBHs specifically, but do black holes obey the No Hair Theorem? (Or, put another way, is GR correct in the strong gravity regime?) This is being tested with the Event Horizon telescope, which is attempting to observe the event horizon of the SMBH at the center of the Milky Way.
They're fascinating objects, and due to their enormous size and energetics they are relevant to a huge number of problems in astrophysics and fundamental physics.