This usage of the word water seems to have gone back and forth a bit between singular and plural.
The second edition of the OED from 1989 gave this definition for water (sense 17.c):
The fluid contained in the amniotic cavity (liquor amnii); now usually pl. The effusion of this fluid from the womb, which precedes the exclusion of the fœtus, is popularly denoted by the expression ‘the waters have broken’. [My emphasis]
Note how it says “now usually pl”, rather than just “usually pl”. The first citation is from 1688 and uses the singular, whereas the other two citations (there are only three in all) are from 1754–64 and 1880 and both use the plural, though they also do not involve the verb break.
The third edition of the OED from September 2015 has changed this definition of the word (which is now sense 19):
Amniotic fluid. In later use freq. in pl.
Freq. as the subject or object of the verb break, with reference to the rupture of the fetal membranes and release of amniotic fluid during delivery of a baby (or the young of an animal).
More citations have been added as well. There are now ten citations in total, of which six use the singular and four the plural. Three citations use the word as the subject of break (of which one, from 1958, is plural and two, from 1658 and 1991, are singular); one, from 2005, uses it as the object in the plural (“When the registrar broke my waters”). The rest don’t use it with the verb break at all. All the citations that use the plural are British, and the ones that use the singular are American.
Google Ngrams are perhaps not really the most useful tool for this expression, since both versions (“Her water broke” and “Her waters broke”) almost flatline completely until the 1940s—presumably such a matter was too delicate to be entrusted quite so directly to paper in earlier times. Really usable statistics seem to start around 1960 or so. If you limit your search to the period from 1960 onwards, it is clear that the singular has been far more common than the plural overall. If you break down your search by corpus, however, you’ll see that there is a clear dialectal difference here:
In British English alone, the two were fairly neck-and-neck (with data points that look somewhat unreliable) until around 1980, after which the plural took hold and the singular began to wane. The plural is now about twice as common as the singular.
In American English alone, on the other hand, not only has the singular been more common throughout, since the 1960s, it is also now more than ten times as common as the plural.
So for current English, the simplistic answer seems to be that the singular is American and the plural is British.