The phrase (as) scarce as hen’s teeth, and its variants, mean very rare.

—Cf. also the phrases as scarce as rocking-horse manure and as scarce as pork chops in a Jewish boarding house.

The phrase (as) scarce as hen’s teeth occurs, for example, in Why am I so furious about teeth? They are deeply socially divisive, by Emma Beddington, published in The Guardian (London and Manchester, England) of Tuesday 25^th May 2021:

What makes me angriest about teeth is how deeply socially divisive they are. Our idiotic enamel lumps are created more or less equal, but after that, dental outcomes depend almost entirely on income. Dentistry is expensive, and NHS dentists as rare as, well, hens’ teeth (I joined an NHS waiting list in 2018; three years on, they occasionally email to tell me nothing has changed).

The expression hen’s teeth has long been used as a type of something which is extremely rare, unattainable or non-existent.

In early use, the expression hen’s teeth denoted an imaginary item that a gullible person may be sent to find as a joke. The earliest occurrences of this expression that I have found are as follows:

1-: From Amusements Serious and Comical, Calculated for the Meridian of London (London: Printed for John Nutt, 1700), by the English translator and satirist Thomas Brown (1663-1704):

Hark you, Mr. Broker, I have a Parcel of excellent Log-Wood, Block-Tin, Spiders Brains, Philosophers Guts, Don Quixot’s Windmills, Hens-Teeth, Ell-Broad Pack-Thread, and the Quintescence of the Blue of Plumbs ¹. Go you Puppy, you are fit to be a Broker, and don’t know that the Greshamites ² buy up all these Rarities by Wholesale all the Year, and Retail them out to the Society every first of April.

¹ This is probably a variant of the blue of the plum, denoting the bloom of youth; freshness, charm.
² From the name of Gresham College, one of the early meeting places of the Royal Society, the noun Greshamite, also Greshamist, designated a fellow of the Royal Society. Gresham College was founded by Thomas Gresham (c. 1518-1579), English mercer, merchant adventurer and founder of the Royal Exchange.

2-: From Poor Robin’s Almanack (London, England) for 1738—as quoted by the English antiquarian and topographer John Brand (1744-1806) in Observations on the Popular Antiquities of Great Britain: Chiefly illustrating the origin of our vulgar and provincial customs, ceremonies, and superstitions (London: Henry G. Bohn, 1849)—the following is about April Fools’ Day:

The following verses on the tricks practised on this day occur in Poor Robin’s Almanack for 1738,—
‘No sooner doth St. All-fools morn approach,
But waggs, e’er Phebus mount his gilded coach,
In sholes assemble to employ their sense,
In sending fools to get intelligence;
One seeks hen’s teeth, in farthest part of th’ town;
Another pigeons milk; a third a gown,
From stroling coblers stall, left there by chance.’

The earliest occurrences of the phrase (as) scarce as hen’s teeth and variants that I have found are as follows, in chronological order:

1-: From the Boston Patriot & Mercantile Advertiser (Boston, Massachusetts, USA) of Wednesday 10^th August 1831:

Virginia Elections.—The elections for Congress and the State Legislature, take place in each county on the Court days, respectively. A writer in the Richmond Whig says, in relation to the vote for Mr. Newton, in Princess Anne county, that “it plainly shows that the aspiring pretensions of Jacksonism in Princess Anne, are laid in the dust, never more to disturb that peaceable county, with its bluster and humbug. Norfolk County can scarcely be said to have partaken of the infatuation; Nansemond is fast recovering from it, and the Borough has exhibited the most favorable symptoms of returning health, from the dose administered by Judge Berrien. In a word, if there is any virtue in signs, Jacksomen [?] will be as scarce as hens’ teeth in this district two years hence.”

2-: From an unsigned letter to the Editors, published in the Richmond Enquirer (Richmond, Virginia, USA) of Saturday 18^th February 1832:

Poetry, in these days, is not quite as scarce as hen’s teeth—being found even among us in exotic abundance but it cannot be considered an indigenous plant springing in rich luxuriance from the soil of Virginia.

3-: From The Merchants, published in the Vicksburg Register (Vicksburg, Mississippi, USA) of Thursday 10^th April 1834:

If there is a Jackson man among the merchants of Vicksburg, we ask him to give the following extract from the Globe an attentive perusal. We believe that Gen. Jackson’s adherents among the merchants of our city, are almost as scarce a “hen teeth,” and such praise and complments [sic] as the following must render them still scarcer.

4-: From The North Carolina Standard (Raleigh, North Carolina, USA) of Friday 15^th May 1835:

The Great Whig Family.—It is quite lamentable for us to announce the great mortality which has of late fallen upon that aristocratic branch of the body politic, styling itself “the great Whig family.” Some unknown epidemic has killed off immense numbers of that unfortunate race of “nation savers.” A few months ago the were as “thick as blackberries;” now they are almost “as scarce as hen’s teeth.”

5-: From the Fayetteville Observer (Fayetteville, North Carolina, USA) of Tuesday 9^th June 1835:

THE CAUSE IS GOING AHEAD.

We have recent accounts, that can be relied on, from every Western County in the State, from which it appears that Judge White is carrying every thing before him. The Jackson men, in the Western Counties, will not be transferred, like so many horned cattle, to Martin Van Buren, either by the Caucus or the sub-agency at Raleigh, and they are every where rallying under the White flag with astonishing unanimity. […] Now a majority are almost unanimous for White, while Martin Van Buren is weak every where. In Guilford, Davidson, Rowan, Montgomery, Anson, Cabarrus, Wilkes, and Burke, Van Buren men are as scarce as hen’s teeth.

 

A FRENCH PHRASE

 

There is, in French, the phrase quand les poules auront des dents, literally when hens have teeth, which can be translated as never in a month of Sundays.

The earliest occurrence of the phrase quand les poules auront des dents that I have found is from Matinées sénonoises, ou proverbes françois, suivis de leur origine (Paris: Chez Née de la Rochelle – Sens: Chez la veuve Tarbé – 1789), by Jean-Charles-François Tuet (1742-1797)—the following passage occurs after the explanation of calendes grecques (i.e., Greek calends):

Les anciens, pour dire qu’une chose n’arriveroit jamais, disoient encore qu’elle arriveroit quand la mule auroit mis bas, quum mula pepererit. Les Picards disent, quand les poules auront des dents.
     translation:
The Ancients, to say that something would never happen, also said that it would happen when the mule would have given birth, quum mula pepererit. The Picards say, when hens have teeth.