In the airlines, the number of flights where an emergency has been declared is about 1 in 600. In general aviation, it could be as low as 1 in 40 or as high as 1 in 6,000, but it's not "most".
In the airlines:
Poking around Google gives this USA Today story: Medical emergencies occur on 1 of every 604 flights, which references an article in the New England Journal of Medicine.
An answer on a different question on this site provides the following anecdote:
had occasion about a year ago to ask our airline safety department
what the most common cause was for declared emergencies in our
operation, and was told that far & away, sick passengers (medical
emergencies) were #1 by a huge margin, and everything else (which
includes all the aircraft-related events) was far down in the weeds in
We'll assume that Ralph J's answer is correct and that medical emergencies are the lion's share of total emergencies. That being the case, the 1 in 600 number is probably accurate within an order of magnitude.
In general aviation:
Since 2001, the NTSB aviation accident database has been remarkably consistent in terms of the number of reports they get:
- About 4,000 reports per two-year span (until 2013, when it dropped to a little under 3,500)
- Between 60-90 reports containing the phrase "declared an emergency" within that span
As airline travel continues to be outrageously, shockingly safe, let's run with the assumption that most accidents in the database are non-airline accidents. That being the case, one in 30-40 of accident airplanes (not all airplanes) declared an emergency in a way that managed to make it into the accident report.
That doesn't fit with a fact I found in an AOPA handout though, which says that air traffic controllers provide more than one flight assist per day. "More than" one could be anything, but let's be super generous and say there are 1,000 flight assists per year. Another FAA source says that there were 24 million GA flight hours in 2015. Figure a GA aircraft holds about 4 hours of fuel, then there were around 6 million GA flights in 2015, or one flight assist per 6,000 flights. That throws a wrench into our other estimate, but it also goes against the "very rare" statement of the inspector in the original question.
From personal experience
Anecdotally, I've flown a few thousand hours and I would say that I've heard aircraft on the radio requesting emergency services less than a dozen times. If I had to guess, I would put the number closer to the 6,000 side than the 40 side, but even if 1 in 40 was correct, 39 pilots out of 40 not having declared an emergency seems to contradict the statement of the inspector.
The TC inspector could be suffering from selection bias. If she or he deals with emergencies all day, it seems less likely that she would come into contact with the (seeming) majority of pilots that haven't declared.