I'm considering moving to the US or Canada and know that I will almost definitely have to buy a car to live there, but, for reasons I needn't expound, I feel extremely unsafe driving a car with an automatic transmission; How prevalent are manual transmissions on the quality used-car market? — I know that at least in the late '90s you occasionally would still see "normal" stick-shifts (i.e. like the car below and not trucks, sportscars or tanks), but I'm afraid that cheap, reliable clutch-based transport may have gone the way of the dodo.
In the US, not so easy to find except for sports cars or very low end. You will not find a US high or middle market sedan or SUV with a manual transmission. Having said that, American, Japanese and Korean automatic transmissions work very well and are completely safe.
In general it's not a problem to find a manual car here in Canada. 8 years ago When I came over from Europe I got a used Hyundai Elantra. Kijiji lists ~95k automatic and ~15k manual cars for sale. But - my wife can't drive it, many other people can't drive manual. So you'll always be the designated driver :-). Pricewise it's not cheaper, maybe even more expensive than an automatic. Today's automatic seem also to be more fuel efficient than a manual car. That was not the case in the past, but due to the recent development in electronics it's more difficult to beat the automatic. Have fun driving around here. The signs are all different, the signal (if they signal at all) is a red light not a yellow one. There is no mandatory safety check on cars on a regular basis, so even the oldest, unsafest cars can still be driven. And speeding can be very expensive :-).
I haven't looked at the used car market in a long time but when I was car shopping in 2013 I started out looking at manuals. Everybody I looked at had some but only at the low end of the scale and what they had tended to be rather bare bones also.
So long as you are looking at the cheap end I think you'll find something. More expensive cars, you very well might not.
New car? Several models still come with manuals, including some very decent cars, e.g. the 2018 Honda Accord. Used? You'll have to do some digging, but they're out there. (I still drive a loaded 2007 Accord EX/navi with a 5MT, that I bought new in '07.) One good thing is that if you can play coy about preferring a manual, you can often get a decent deal on one, since relatively few people in North American can drive them.
I think they're somewhat more popular in Canada than in the US, but they are not particularly popular anymore in either country. But they are out there.
In the US can buy used cars directly from the seller, from new car dealerships, used car dealerships, and from online marketplaces like autotrader.com (no affiliation). One of the nice things about the online marketplaces is you can search based on a number of criteria. For example, looking at used sedans under 20k USD available on autotrader.com (and my guess is that any other used car marketplace offers a similar selection) in a number of major cities, there are lots of manual transmissions available, but they tend to be more sporty than the ones with automatic transmissions.