Querying non-ASCII rows from Postgres

To answer your question: [:ascii:] works. You may have some characters in your text that you do not recognize as non-ASCII, yet they're there. They can be something like a non-breakable space, for instance, or any other Unicode space character.

It is not strange to have non-breakable spaces ( ) in texts that you cut-and-paste from a web page, yet you don't notice they're there.

Here is an example to show:

WITH t(t) AS
(
    VALUES 
      ( 'Сталинская правозащитница: мать Меленкова бабушка Настя' ),
      ( 'Дневник НКВДиста Шабалина: Знает ли Москва положение на фронте?' ),
      ( 'Бег по городу и поездка на осле: как в средневековье наказывали прелюбодеев' ),
      ( 'Как комиссар Крекшин в 1740-е чуть не отменил историю России' ),
      ( 'Have you heard of Saint Death? Don’t pray to her.' ),
      ( 'Архаїчна українська мова: перевага чи недолік?' ),
      ( 'Гренада не их' ),
      ( 'China’s marriage rate is plummeting because women are choosing autonomy over ' )

)
SELECT 
    t,  regexp_replace(t, '([^[:ascii:]])', '[\1]', 'g') AS t_marked
FROM 
    t 
WHERE 
    t ~ '[^[:ascii:]]' ;

That's what you get:

                                       t                                       |                                                                                                 t_marked                                                                                                  
-------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
 Сталинская правозащитница: мать Меленкова бабушка Настя                       | [С][т][а][л][и][н][с][к][а][я] [п][р][а][в][о][з][а][щ][и][т][н][и][ц][а]: [м][а][т][ь] [М][е][л][е][н][к][о][в][а] [б][а][б][у][ш][к][а] [Н][а][с][т][я]
 Дневник НКВДиста Шабалина: Знает ли Москва положение на фронте?               | [Д][н][е][в][н][и][к] [Н][К][В][Д][и][с][т][а] [Ш][а][б][а][л][и][н][а]: [З][н][а][е][т] [л][и] [М][о][с][к][в][а] [п][о][л][о][ж][е][н][и][е] [н][а] [ф][р][о][н][т][е]?
 Бег по городу и поездка на осле: как в средневековье наказывали прелюбодеев   | [Б][е][г] [п][о] [г][о][р][о][д][у] [и] [п][о][е][з][д][к][а] [н][а] [о][с][л][е]: [к][а][к] [в] [с][р][е][д][н][е][в][е][к][о][в][ь][е] [н][а][к][а][з][ы][в][а][л][и] [п][р][е][л][ю][б][о][д][е][е][в]
 Как комиссар Крекшин в 1740-е чуть не отменил историю России                  | [К][а][к] [к][о][м][и][с][с][а][р] [К][р][е][к][ш][и][н] [в] 1740-[е] [ч][у][т][ь] [н][е] [о][т][м][е][н][и][л] [и][с][т][о][р][и][ю] [Р][о][с][с][и][и]
 Have you heard of Saint Death? Don’t pray to her.                             | Have you heard of Saint Death? Don[’]t pray to her.
 Архаїчна українська мова: перевага чи недолік?                                | [А][р][х][а][ї][ч][н][а] [у][к][р][а][ї][н][с][ь][к][а] [м][о][в][а]: [п][е][р][е][в][а][г][а] [ч][и] [н][е][д][о][л][і][к]?
 Гренада не их                                                                 | [Г][р][е][н][а][д][а] [н][е] [и][х]
 China’s marriage rate is plummeting because women are choosing autonomy over  | China[’]s marriage rate is plummeting because women are choosing autonomy over 

You can see from this, that your problem is the right-apostrophe character. ASCII only supports the apostrophe, left-apostrophe and right-apostrophe are typographically correct Unicode extensions

dbfiddle here

You can check it also with previous versions at http://rextester.com/UKIQ48014 (PostgreSQL 9.5) and http://sqlfiddle.com/#!15/4c563/1/0 (PostgreSQL 9.3)

The texts that I guess you think are pure ASCII, and are not:

 WITH t(t) AS
 (
     VALUES 
       ('A fully ASCII text!'),
       ('Have you heard of Saint Death? Don’t pray to her.'),
       ('China’s marriage rate is plummeting because women are choosing autonomy over ')
 )
 SELECT 
    regexp_replace(t, '([^[:ascii:]])', '[\1]', 'g') AS t_marked
 FROM 
    t 
 WHERE 
    t ~ '[^[:ascii:]]' ;

 | t_marked                                                                        |
 | :------------------------------------------------------------------------------ |
 | Have you heard of Saint Death? Don[’]t pray to her.                             |
 | China[’]s marriage rate is plummeting because women are choosing autonomy over  |
 

dbfiddle here

These texts are using ’ instead of ' to mark apostrophes.

Check Punctuation: Why is the right single quote (U+2019), and not the semantically distinct apostrophe (U+0027), the preferred apostrophe character in Unicode? ... to see that you're not the first person encountering this problem.