I have heard all three of these expressions in various parts of the US to describe the disappearance of things.
All three expressions appear to be readily understood. Are some more common in certain regions of the US or in other parts of the English speaking world?
Addendum: Growing up in Texas, I most commonly heard to take missing, which I suppose arose analogously from to take sick.
In movies and TV, to go missing is more common and presumably the standard form of the expression.
But one sometimes hears the contradictory to turn up missing.
To turn up missing is not a contradiction because here to turn up X with a complement means something completely different from the similar to turn up that means to put in an appearance. To turn up X instead means to turn out to be X or (to be) found out to be X. It does not mean that it made an appearance. These are different constructions with different histories.
See also to turn up trumps for a phrase that works like to turn up missing and not like to turn up or to turn up something.
to go missing
The phrase to go X, where X is an adjective or prepositional phrase, is an construction handed down to us from time immemorial; for example, Old English had to go on two meaning to be or become divided in two, and Middle English sees Chaucer having people going out of their wits for woe in the Knight’s Tale.
Per the paywalled OED, the general construction means:
intr. To pass into a certain state or condition (often implying deterioration).
a. With adjective or prepositional phrase as complement. To pass into or come to be in a specified state or condition; to become.
Often in phrases indicating mental decline, as to go out of one’s mind, etc. to go crazy, to go loco, to go mad, to go off one’s rocker, etc.: see the final element.
It then mentions many phrases like going bankrupt, going commando, going rogue, going native, going public, and going missing. About the latter the OED say:
to go missing: to become lost or not able to be found, esp. through not being in an expected place.
The earliest citation provided is from the New York Herald (U.S.) in 1845, but the latest citation is from the Carmarthen Journal (Wales) in 2011. There’s also a nineteenth-century Australian citation, so I don’t believe this is limited to any one corner of the anglosphere. It is not marked casual or colloquial.
to turn up missing
The intransitive phrasal verb to turn up of something appearing casually or unexpectedly, or to arrive or present oneself but not unexpectedly, doesn’t appear until the 1700s. When used with a complement, it means per the OED:
To turn up trumps meaning to turn out well or successfully has been around since the 1500s. You can find plenty of examples of it in Google Books (although you need to discard the instances talking about card games).
The three citations for to turn up X that the OED provides shows you that the sense here really is nearly equal to that of to turn out to be X:
- 1756 Monitor No. 39. I. 374 A great deal of waste land and timber.., which by care and cultivation, must in time turn up a great thing.
- 1831 Examiner 534/1 A lottery ticket which has turned up a prize.
- 1872 Judy 29 May 59/2 (Farmer) Have the ornaments [= handcuffs] handy, in case he should turn up rough.
So to turn out X really means to turn out to be X. It doesn’t mean that it showed up.
to take missing
I have not been able to uncover any documentation or examples of to take missing in the sense of to go missing. That suggests this sense may be a regional one restricted to a small corner of the anglosphere and little known without.