How common is the term Ho Chi Minh City outside of official circles?
In the South, not common at all; though I do believe people in the North tend to refer to the city with the official name.
If you ask any Saigonese, most will say they much prefer Saigon. Personally, I hate the official name.
Same in the North, they use Saigon all the time.
And even the airport IATA code is SGN.
It's used almost exclusively in official/government contexts, on road signs, in organization names, etc. Often shortened to HCM or "tp HCM" (thanh pho)
This exercise is the name of a famous song.
IME use of HCMC in the south is fairly common. Sai Gon is much more common though.
Đà Nẵng, Nha Trang, Cần Thơ, Hải Phòng, Đà Lạt...
Saigon’s name may or may not be Chinese.
Also, note that in Saigon itself, the name would be pronounced with a Russian-style sh-sound (ш).
Is that true of the s generally in the South, or generally in front of back vowels?