All the same rules apply to single quotation marks. So when you have a single quote mark followed by a double, that doesn’t change where the period goes. It goes inside all the quote marks.

Double quotation marks are used for direct quotations and titles of compositions such as books, plays, movies, songs, lectures and TV shows. They also can be used to indicate irony and introduce an ...

The printers and publishers who shaped the rules here in America believed that a period or comma after a closing quotation mark looked bad. So they decided it should stay inside, logic be damned ...