Nearly all American copy editors tend to have standing instructions to change which to that if it introduces an integrated relative clause. This is a tragic waste of many hundreds of people's time. It has never been the case that which is forbidden, or uncommon, in integrated relative clauses. It is quite uncommon for ambiguity to arise over whether a relative clause is integrated or supplementary, because flanking commas should always be used to mark off the latter; so insisting that only supplementary relatives are allowed to begin with which does hardly any work in reducing ambiguity.
Here is a list of a few Language Log posts discussing the issue (these were written for fun more than scholarship; the tone, as usual on Language Log, ranges from light-hearted detachment through stern criticism to mock rage):
Sidney Goldberg on NYT grammar: zero for three (Geoff Pullum, 9/17/2004)
Which vs. that: I have numbers (Geoff Pullum, 9/19/2004)
Which vs that: a test of faith (Mark Liberman, 9/20/2004)
Which vs. that: integration gradation (Mark Liberman, 9/23/2004)
Don't do this at home, kiddies! (Arnold Zwicky, 5/3/2005)
The people from the CCGW are here to see you (Arnold Zwicky, 5/7/2005)
What I currently know about which and that (Arnold Zwicky, 5/10/2005)
Five more thoughts on the that rule (Arnold Zwicky, 5/22/2005)
Ann Coulter, grammarian (Mark Liberman, 10/7/2005)
That which doesn't apply to English (Geoff Pullum, 7/3/2010)
Which-hunting in uncomprehending darkness (Mark Liberman, 5/4/2012)
Preaching the incontrovertible to the unconvertible (Geoff Pullum, 12/6/2012)
A decline in which-hunting? (Mark Liberman, 12/6/2012)
Those interested in the statistics on using which vs that in integrated ("restrictive") relative clauses can find some figures in print, in the Longman Grammar of Spoken and Written English by Douglas Biber and colleagues. Page 616 provides some figures for relative clauses in American and British newpapers (in approximate numbers of occurrences per million words):
|AmE news||BrE news|
|integrated relatives with which||800||2600|
|integrated relatives with that||3400||2200|
|supplementary relatives with which||1400||1400|
|supplementary relatives with that||0||0|
The striking figure is the last one: virtually no supplementary relatives are found with initial that any more. Actually such examples do occur occasionally (though the figures above were based on texts that happened not to contain any). Genuine supplementary relatives can be identified unambiguously when the head noun is one that doesn't take integrated relatives at all, like a proper name or a similarly uniquely referring definite NP. So the sentence His heart, that had lifted at the sight of Joanna, had become suddenly heavy is an example of a supplementary relative beginning with that. But such examples are extremely rare. See this Language Log post for the story of one confirmed sighting of this rare species in the wild.
On the choice between that and which in integrated relatives, there is a clear frequency difference between the dialect groups — Americans tend to use which less; but reading a broad range of English texts makes it obvious that both that and which are grammatical in integrated relatives in both dialect groups. Biber et al. speculatively attribute the difference to cultural style: a greater "willingness to use a form with colloquial associations" among Americans. It is not clear that there is really any support for this. It seems more likely to be a response to what so many prescriptive books still (wrongly) tell them.
What is clear is that the prescriptivists are simply wrong about what should be prescribed: which relatives are common, both from Americans (think of President Roosevelt's a day which will live in infamy, uttered the day after the attack on Pearl Harbor) and British writers ("It is the greatest blow which has befallen me in my career", from The Hound of the Baskervilles by Sir Arthur Conan Doyle, or "Never use a metaphor, simile or other figure of speech which you are used to seeing in print" from George Orwell). Thousands of examples from competent and even great writers could be cited from either side of the Atlantic.