This is very simple genetiv. The brain belongs to someone. The house belongs to the mother.

Re your examples:

- “brains” is often used colloquially in the plural (cf. “wits”)

- *"I moved with my mothers" is just wrong, unless you have two mothers. No one would say this.

- *"I moved with my mother's place” is also wrong – there are various possible correct versions of this sentence, such as “I moved to my mother's (place)”, in which case “mother's” is a simple genitive (cf. I buy bread at the baker's) .

"blow your brains out" is a plural - "brains" being the matter in your brain, rather than the entire object. Similar to "guts".

I'm at my mother's, let's go round to John's, get it from the greengrocer's - as you said, it's the possessive and means "my mother's house", "John's place", "the greengrocer's shop".

SO as I now understand it's a colloquially used plural - thaks a lot to all of you!!

@steve:
I thought "I moved with my mothers" was said by the girl pretending to be Nicholas Cage's daughter in the movie 'matchstick men'...

Though it was a little different:

FRANK
Suppose you don't show?
ROY
Take her to her mother's. Then
you drive south and never come
back.
FRANK
What are you going to do?


(screenplay extract)

ROY
Take her to her mother's (house). Then
you drive south and never come
back.


The house is implied but not stated

@Sage N. Fer Get K.S.C., re: SO as I now understand it's a colloquially used plural - thaks a lot to all of you!!

Wrong. I would say (and some backed me up) that it is a colloquially used genetiv (one's and mother's). 'Brains' is plural but you didn't ask for the brains - it isn't suffixed.

sure - with the mother's house... but the brains and guts and wits convinced me...