Resolved questions
When you make "falling out" plural, which word receives the "s"?

I think it would be "times we had a falling out", but with the way that phrased is used in America, it means something final. You wouldn't have another falling out with the same person :)

I could see myself using every one of them, and more possibilities besides.

I think that both are acceptable while I don't use fallings out.

I prefer "fallings out", because it is the fallings which are plural (what are "outs"?). Technically "falling-outs" is also grammatically correct, but I would never use this form because I have an irrational hatred of hyphens.

this question is interesting for me cause it is one wondering about his own native language. Well, an opinion from an non native (linguistically speaking!). Normally, in all languages, are sustantives that can be pluralized. How do you pluralize a preposition?
It all depends on how the composite word has been lexicalized. But refer to standards proposed by any authority on the matter (difficult in your case, cause i think anglosaxon culture abhors of "centralized authorities"). So, maybe falling-outs is ok, but "falling out(s)" no. Or maybe it is good standard ...a pair of falling out, some falling out, a number of falling out, etc. Now, the only cases i can think of correct use is "ins" and "outs", and here "in" and "out" has lost its primary meaning as preposition. Anyway, if you get a reliable info about this, tell me, i ll be grateful. And popular/street use of language is interesting to be noticed (and analized) but nornmally tells nothing about the "standard/normative" grammar of the language.

@ Josep

"So, maybe falling-outs is ok, but "falling out(s)" no."

I'd have to disagree with you there. Both are perfectly fine.