But what I would like to know is, what are the most common emergencies that beyond-novice divers tend to face? What are the highest probability emergency issues one might encounter? And how common are they?
A reasonable set of questions, that are almost impossible to answer absolutely. Part of the challenge is agreeing on what constitutes an 'emergency'. Lynne (TSandM) provided what I consider the most appropriate definition. And, if you accept that characterization, then your last question is the easiest to answer: How common is any emergency? Very, very,
UNCOMMON - rare / virtually unheard of. You have gotten a number of good examples of
problems that might occur during a dive, many of which might / should cause the reasonably prudent diver to terminate the dive, and surface. But, with the possible exception of a true out of air situation, occurring at depth, without a buddy nearby - which is also rare - it is heard to describe them as an emergency. Catastrophic equipment failure is quite rare. And, even when it occurs, if you have properly planned a dive, the chances of even that creating an 'emergency' are quite small. A regulator starts free-flowing, breath off the free flowing second stage. (I tell students that a free-flowing regulator is NOT an emergency, it is an annoyance and an inconvenience.) Your buoyancy system fails, swim the unit to the surface. You get separated from your buddy? Search for one minute and surface (or execute whatever alternate plan you and your buddy have agreed beforehand). Perhaps, the one example of what I would consider an emergency is the situation described by DeputyDan, where a diver in the water is hit in the head by a falling cylinder (and diver) from above them, and rendered unconscious / unresponsive. And, that is also very, very uncommon (although I think about the possibility every time I dive a site like Oil Slick, where a diver is exiting via a ladder ahead of me, and I forcibly remind myself to stay out of the way. Perhaps, the most common 'emergency' is diver panic - not the instigating event (regulator free-flow, entanglement, etc), but the diver's reaction to the event. While I subjectively believe that panic is more common than catastrophic equipment failure, or gas management problems, or entanglement, I have no data to say how often it actually occurs.
In 10+ years and 1000+ dives, I have never personally experienced any situation that I considered an 'emergency'. I have witnessed a diver die, after they apparently became separated from their instructor on a technical training dive, and were subsequently found unresponsive on the bottom at 165 feet. But, what exactly happened was never determined, and it is hard to say what the 'emergency' was. Perhaps, that was a matter of panic, but it was impossible to determine if that was the case. I have witnessed a diver bolt to the surface from 50 feet, after getting water in their mask, and up their nose. That is a situation that I considered to be panic. The outcome was psychologically negative, although there was no physical harm.
The main comment essentially was, if you prepare properly, any emergency will be unlikely.
And, that guidance is probably more helpful than any summary of emergencies, and their numerical probabilities, that we could offer.