In British and Australian English, the intransitive verb meaning to place oneself horizontally is “to lie”. In American English, “to lay” is usually used.
("To lay” is used in British and Australian English transitively, e.g. I lay the table, and of course “lay” is the past form of “to lie” in this sense.)

However, I’ve noticed that sometimes “to lie” is also used in American English, in this way. I’m not sure whether this is a regional/individual thing, or whether an individual might use either term in different situations, perhaps due to some shade of difference in meaning: as an outsider I can’t tell how the words “feel” inside the heads of speakers of US English :-) . What do you reckon?

They don’t lay the table in the States? What verb would be used?

They do lay the table in the States. In saying that “to lay” is used in British and Australian English transitively, I wasn’t meaning to imply that it wasn’t used transitively in the USA.

EDIT: perhaps I should have said “used only transitively in British and Australian English, to make my intended emphasis more plain.

EDIT: transitively and reflexively, I suppose ... someone might say “I’ll lay myself at your feet”.

My experience with this verb has mostly been in the context of doctor’s examinations and I would say that lay is used intransitively by the vast majority of nurses.  I get told to “lay down on the table” far more often than I get told to “lie down.” This seems to be part of a general move in AmerE of lay taking over the job of lie.  As a side issue, while we will occasionally “lay the table” we will more often “set the table.” At least, that is what I hear.  Others may hear “lay the table” more often than I have.

Lay/lie may be the most commented on and written about usage item of all time.

First, I’m not aware of any regional distinction. As far as I know, the situation in Britain and North America are the same, although most of my sources on the subject are American.

As usual, Merriam-Webster’s Dictionary of English Usage gives the most thorough review.

The intransitive lay has been going strong since the late thirteenth century. (e.g., from the late 13C poem The Harrowing of Hell: “Sathanas, y bynde the, her shalt thou lay.") Then, as is often the case, eighteenth century grammarians started to declare it incorrect for no particular reason. In this case, the grammarians were rather successful at driving it out of written use, but it has remained very common in speech. M-W notes that it is not a reliable social marker—those who avoid it in writing commonly use it in speech—even though some commentators and writers try to use it as class indicator.

In the late twentieth century, prescriptivists invented another distinction, that the intransitive lay may be used for objects, e.g., “the book lays flat,” while lie should be used for people. Again, this distinction does not seem to be made in actual usage.

Also, in the late twentieth century, the intransitive lay started to appear more in published writing, indicating a loosening of the shibboleth. But it’s still not completely accepted by copy editors. Most grammarians, even if they like the distinction, have declared that attempting to maintain it is futile. Garner rates it a 4 on scale of language change, “virtually universal but is opposed on cogent grounds by a few linguistic stalwarts (die-hard snoots).” (The difference between stage 4 and stage 5 appears to be Garner’s idiosyncratic preference. If he personally observes a widely ignored rule, he and those like him are “linguistic stalwarts” and it rates a 4. If he doesn’t, those who insist on the distinction are “pseudo-snoot eccentrics” and it rates a 5. In Garner’s parlance, being a snoot is a praiseworthy thing.)

The intransitive lay is also used in nautical speech, where it has always been uncontroversial and considered standard, e.g., “lay at anchor.”

As far as I know, the situation in Britain and North America are the same, although most of my sources on the subject are American.

Well I suppose I don’t have any hard data on it to hand. In Australia it would be considered an odd mistake whereas in the US it is an accepted variant: that’s not prescriptivism, just description. Not sure how much influence 18th century grammarians had in Australia since the population was mainly illiterate convicts but somewhere along the way it must have dropped out.

In the late twentieth century, prescriptivists invented another distinction, that the intransitive lay may be used for objects, e.g., “the book lays flat,” while lie should be used for people. Again, this distinction does not seem to be made in actual usage.

That’s a new one to me.

Here is a non-American data point. The OED, in a 1989 second edition entry—but in a note that I’m thinking is an older relic because no OED editor of recent vintage would ever write something like this—says of the intransitive lay: “it is only dialectal or an illiterate substitute for lie.” (When the OED says dialectal, the editors usually are referring to dialects within Britain.)

And as for usage down under, the 2007 Cambridge Guide to Australian English Usage makes the lie = intransitive / lay = transitive distinction, but says:

But the common colloquial trend is to use lay (and laid) instead of lie (and lay/lain).

And:

All these replacements are used in common talk in all English-speaking countries, though in the written medium (certainly in edited writing) the standard forms lie/lay (lain) are still expected. We may speculate on when the pressure of usage will allow their replacements (lay/laid) to prevail in writing; but for the moment they remain markers of informal speech. In the longer run they spell the doom of [intransitive] lie.

(shrugs) I keep my ears open, and to hear an Australian say that would clang.

I know we set little store in Googlematches here but when results are as striking as the following it probably means something

“lying in the road” About 13,600,000 results
“laying in the road” About 2,490,000 results
“lying in the road” site:au About 572,000 results
“laying in the road” site:au 8 results

EDIT: pardon me, I should not be so cocksure. I’m just basing this on my own experiences and impressions which may be atypical or biased.

Yes, this is a case where you should question your own experience. Virtually every professional language resource agrees on the state of affairs regarding lie/lay, although they may differ in their normative judgments.

And it seems likely, especially given Australia’s convict heritage, that the usage has been in common parlance down under all along. The educated classes who did most of the writing would be aware of and adopt the formal prescriptions of British speech.

And of course none of this means that people don’t make the distinction in speech. It’s just that ignoring the distinction has also always been common and standard in the informal register. I’ll bet you’ll start hearing it all the time now that you’re attuned to it.

The difference in the general Google numbers is probably accounted for by the difference in edited v. unedited writing, especially given the penchant of newspapers to reprint one another’s stories. If a paper prints an article, which being edited prose will probably make the transitive/intransitive distinction, it is likely to be reprinted some half dozen times by other papers and websites, resulting in multiple Google hits for a single instance. But you’re right, the Australian numbers are striking. But I’m inclined to suspect that there is something wrong with Google’s search algorithm here. The difference is just too striking to be real. I can’t believe there are less than ten instances in the entire Australian contribution to the internet, no matter how rigorously the rule might be observed there.

The intransitive lay is also used in nautical speech, where it has always been uncontroversial and considered standard, e.g., “lay at anchor.”

Do you mean uncontroversial and considered standard in Leftpondia, Rightpondia or both, DW? Because to my Rightpondian ears that phrase only makes sense in the past tense (’that day our ship lay at anchor in the bay’).

“and we jolly sailor boys were laying up aloft, and the landlubbers laying down below”

there are many versions of this song. Several of them say “lying”, not “laying”. The version I learned as a boy is given above.

I think Patrick O’brian uses the expression “lay aloft”, not “ lie aloft”. I’ll have to try and check that one. O’brian is reputed to be very sound on nautical terms. I wouldn’t know about the reliability of anthologists of songs.

Is there a seagoing layer/lier/liar in the house?

in the US it is an accepted variant

This is not true.  It is common, but it is not “accepted” in the sense I’m pretty sure you mean.