This reminds me of something from "1066 and all that", a gentle parody of history teaching in British schools by W. C. Sellar and R. J. Yeatman, published in 1930.

"Noticing some fair-haired children in the slave market one morning, Pope Gregory, the memorable Pope, said (in Latin), 'What are those ?' and on being told that they were Angels, made the memorable joke — 'Non Angli, sed Angeli' ('not Angels, but Anglicans') and commanded one of his Saints called St Augustine to go and convert the rest."

What the pope was told,of course, was that the children were Angles, and is reply was "Not Angles, but angels" - which perhaps gives us a clue where Anglo comes from - Angli, the medieval Latin name for the English .

Another "1066 and all that" translation: for "Honi soit qui mal y pense"- the motto on the Order of the Garter - they have "Honey, your silk stocking's hanging down".

I think it's mainly due to one ("use your brain") being a simple description, and the other ("rack your brains") being an idiom. But no doubt there has been a bit of cross-fertilisation between the two.

In both American and British books, "use my brain" is more popular than the plural version.

But with "rack" it's a bit different. The idiom often seems to be listed in dictionaries etc as "rack your brains" (although both are allowed):

http://www.phrases.org.uk/meanings/rack-your-brains.html

However, while in British English "rack my brains" is still more popular than the singular version, in American books the singular seems to have overtaken the previously more popular plural version:

http://books.google.com/ngrams/graph?content=rack+my+brain%3Aeng_us_2012%2Crack+my+brains%3Aeng_us_2012%2Crack+my+brain%3Aeng_gb_2012%2Crack+my+brains%3Aeng_gb_2012&year_start=1800&year_end=2000&corpus=15&smoothing=3&share=&direct_url=t1%3B%2Crack%20my%20brain%3Aeng_us_2012%3B%2Cc0%3B.t1%3B%2Crack%20my%20brains%3Aeng_us_2012%3B%2Cc0%3B.t1%3B%2Crack%20my%20brain%3Aeng_gb_2012%3B%2Cc0%3B.t1%3B%2Crack%20my%20brains%3Aeng_gb_2012%3B%2Cc0

At Google Books I can find no "rack" examples before 1700, but between 1700 and 1750 there are 12 examples each for "brains" and "brain", although in the second half of the 18th century "rack my brain" became more popular. It seems that the "brains" version became more popular in the 20th century - 61- 46 examples at Google Books.

As to your final question I don't think they have any different meaning, it's simply a matter of idiom - "Use your brain" sounds more idiomatic to me, but conversely, so does "rack your brains".

Incidentally, there's another discussion going on about "rack your brains" or "wrack your brains" (Oxford allows both) at DailyWritingTips, StackExchange, and WorldWideWords.

It's a genuinely interesting question, this. It seems to me that in wh-questions with the verb "be" plus a noun, the verb is generally governed by by the noun that follows it: perhaps because it's a subject complement rather than a direct object, it's seen as the subject.

"What are oranges?" (oranges are what?)
"What kind of fruit are oranges?" (oranges are what kind of fruit?)
"What fraction of the fruit are oranges?" (oranges are what fraction of the fruit?)

But judging by searching the Internet, this last type of question is pretty rare. Similar constructions would be (with Google Search figures):

"What percentage of the population are women?" - 9950 ("is" - 5) - women are what percentage
"What proportion of the population are geniuses? "- 7 ("is" - 0) - geniuses are what proportion

When verbs other than "be" are used, it varies, depending on what is seen as the subject:

"What percentage of the population have green eyes?" - 1120 (has - 6680) - "percentage" is the subject of the verb "have"

"What family of fruit do oranges belong to?" - "oranges" are the subject of the verb "belong"

When adjectives are involved after "be", it seems to depend on the noun just before "be"

"What percentage of the fruit is rotten?" (fruit - uncountable)
"What percentage of the people are unemployed?" (people - plural)

Googling "what fraction of the fruit" came up with 61 results. I discounted 3 as being ungrammatical or this thread, 1 was followed by an adjective, 6 were followed by verbs other then "be", and in 6 "fruit" was followed by another word such as "salad" or "slices". That left 42 where "be" was followed by a plural, but of these, 5 were in verb forms which do not differentiate between singular and plural. Of the rest:

"What fraction of the fruit (below/in the bowl) are/were (the) oranges/apples/pears etc?" - 26
"What fraction of the fruit (in the bowl/bag/Dylan ate) is oranges/apples/pears? " - 11

So it seems that, like me, people generally prefer the plural. The singular sounds very weird to me. I have to say, though, that the picture is rather different at Google Books, where singular outnumbers plural 12 - 6. And a respectable publisher can be found for each - Barrons Psat/Nmsqt practice tests- "is", Everyday Mathematics, University of Chicago School Mathematics Project -"are".

There were a couple in Google Search I thought were rather less grammatically ambiguous than "What fraction of the fruit are/is" - one where fraction really is the subject, and one where it is the object:

"What fraction of the fruit salad was made up of strawberries"
"What fraction of the fruit bar did each child get?"

"One" refers to the real subject - "a (highly unusual) form of melody", which is a noun phrase. Take away the pronoun and you have - "It is a highly unusual form of melody that occurs only in this composer’s work".

"I frequently hear the rule that the referent has to be the prior proximate noun. " - perhaps "the prior proximate noun phrase" would be better, especially where 'of' id concerned, although perhaps without the modifiers:

"Can you bring me that box of rather delicious chocolates, the one on the kitchen table?"

"Just look at those friends of David's, the ones hanging around the school gates."

@jayles (the ?) - another for your collection - "a fraction of the people present were/was", etc - see http://painintheenglish.com/case/5237/

Just noticed this, which is an interesting parallel to what we've been discussing elsewhere (the thread on team names) - For a start I would prefer "What fraction" to "Which fraction", but the real question is about verb agreement. We're back to notional agreement vs formal agreement - "A large number of people have recently debated this point" or "A large number of people has recently debated this point"? Me, I go for plural.

Google Search:
"What fraction of the fruit are" - 35
"What fraction of the fruit is" - 15
Note that "Which fraction" gets two hits, apart from this thread.

At Google Books, however, it's 7 to 4. But those 4 do include one from the School Mathematics Project at Chicago University.

'Is' may be grammatically more 'correct', but I find 'are' more natural. Perhaps the problem is that there are two possible full answers to "What fraction of the fruit is/are oranges?" - "Oranges are one third of the fruit" (not "is") or "The fraction is one third of the fruit" (not "are") . My first reaction would be to answer "Oranges are ..."This is because the question sounds to me more like "What fraction of the fruit do oranges comprise?" rather than "What fraction of the fruit consists of oranges?"

It's interesting to see what happens in normal statements (rather than strangely worded questions). I looked up the first 50 entries for "only a small fraction of" at Google Books. There were nine examples where it was the subject and related to a countable noun, and where the verb could be distinguished as singular or plural. The result was 7 plural verbs to only 2 singular ones:

"only a small fraction of the transmitted electrons pass through the objective aperture" (pl)
"only a small fraction of RyRs are activated" (pl)
"Only a small fraction of these cells, however, are able to create a micrometastasis" (pl)
"only a small fraction of the planetesimals originally in the vicinity of Neptune was scattered outward: " (sg)
"If only a small fraction of girls are biologically oriented to market rather than household activities" (pl)
"Only a small fraction of natural deaths receives a forensic investigation" (sg)
"Historically only a small fraction of the benefits from innovations have been captured by the innovator" (pl)
"Yet only a small fraction of America's law enforcement officers have been trained to recognize and stop this serious crime" (pl)
"only a small fraction of auto engines that are not functioning optimally produce the majority of pollutants" (pl)

It occurred to me that social media might be a good way of gauging normal (not published) use. The first figures are for actual instances, the ones in brackets the numbers given on the first search page. Either way they're pretty overwhelming:

"Arsenal have won the"
Facebook 261 (185,000)
Twitter 569 (94,500)
Yahoo Answers 43

"Arsenal has won the"
Facebook 41 (8,750)
Twitter 58 (545)
Yahoo Answers 13

My expanded take on the history of 'early doors', illustrated with lots of examples from the past and present - http://random-idea-english.blogspot.com/2014/06/random-thoughts-on-early-doors.html

I wonder where this figure of 95% comes from; it certainly doesn't seem to apply to British English and like njtt I don't think I've ever heard "I wonder that" used instead of "I wonder if/whether". Perhaps it's a regional thing. These are all the instances of the three variations at the British National Corpus:

"I wonder if" - 840
"I wonder whether" - 126
"I wonder that" - 7 - all of which have the meaning "I'm surprised that" pointed out by njtt

I make that precisely 0%

At Netspeak, based on a web corpus (ie mainly American), out of all "I wonder *" possibilities:
"I wonder if" gets 40.5%,
"I wonder whether" - 0.7%
"I wonder that" - doesn't even register, meaning less than 0.0%

Google Books show over 63 million for "I wonder if", 8 million for "I wonder whether" and just under 2 million for "I wonder that",with most on the first page looking to be valid uses of "that"

Facebook - "I wonder if" - 28.4 million, "I wonder whether" - 199,000. Admittedly "I wonder that" has 915,000, but a glance at the first page suggests that many of these are non-native speakers, and in any case, that would still be only about 3% max.

Twitter - "I wonder if" - 74 million, "I wonder whether" - 1 million, "I wonder that" 473,000

(All Google Search figures are front page figures, which aren't particularly accurate, but should do for comparison purposes).

And remember that all these searches for "I wonder that" includes things like "I wonder that, too", as well as the "surprised" meaning. Substitutions for "if/whether" certainly exist, but appear to be relatively few.

So I don't think we need to bother too much arguing about common usage on this one, as common usage it obviously ain't.

@HS - But I thank you; I'd never heard of "early doors" before, but it looks as though it's now moving outside sporting circles. "Watching on", incidentally, was apparently coined by Jonathan Pearce of BBC Radio 5 Live. Another one for your collection - "a big ask".

A quick look at Google Search suggests that "to contest for the ball" is primarily an Australian expression. As far as I can see, all first ten instances of "contest for the ball" on Google where contest is a verb are Australian, and the expression seems to be used in football, rugby and Australian rules football alike.