As seen recently through the press, Trump claims "North Korea best not make any more threats". What kind of structure and use is the bit "best not make"? Is this informal but acceptable in the US? Is this archaic? Some alternatives sound more acceptable to me, like "should not make" or "had better not to make."
Yes, this is informal and American. It's short for "had best". Here's what OED says:
colloq. (orig. U.S.). An invariable modal verb, normally complemented by the bare infinitive.
Had best; should.
It's been around for over 100 years, but it's not archaic or obsolete.
I live in the Southeastern US, and to me this structure sounds typical of a southern accent. As far as I know, it is always used in a threatening manner. In fact the image I personally have when thinking about this phrase is someone's mother threatening punishment.
In my experience it is usually used alone, where its meaning is understood from context. For instance a child being caught trying to leave when he has chores would be threatened with "You best not!" Particularly frightening because the punishment is left to the imagination!