Bearded elves

[Feanaro]

apart from mahtan which other elf had a beard

[jammi567]

Cirdan, by the end of the third age. And that's it.

[Peter_20]

How can we actually be sure that Elves did NOT grow beards?
Tolkien never stated whether or not they did.

[Landroval]

Tolkien never stated whether or not they did.
Tolkien entertained two different positions; in a note to the Shibboleth of Feanor, HoME XII, he states that:
Elves did not have beards until they entered their third cycle of life
However, he also states in Unfinished Tales:
In a note written in December 1972 or later, and among the last writings of my father's on the subject of Middle-earth, there is a discussion of the Elvish strain in Men, as to its being observable in the beardlessness of those who were so descended (it was a characteristic of all Elves to be beardless)
Make what you will of it :)

[Alcuin]

Fëanor’s father-in-law, Mahtan, had a beard earlier in life than most Elves, and it was apparently reddish like the rest of his hair.

Elves seemed to have aged just as Men, but at a far slower pace. They also seemed to have aged faster in Middle-earth than in Valinor: CÃ*rdan was quite likely the oldest Elf still in Middle-earth, or at least the oldest mentioned in the tales. But even the Valar “aged,â€� as it were (Morgoth’s Ring, “Myths Transformedâ€�):The Valar ‘fade’ and become more impotent, precisely in proportion as the shape and constitution of things becomes more defined and settled.

[Ingwe]

The Valar themselves to fade, that is interesting! In essence, in a scientific and universal way, almost sounds like proton decay and the eventual collapse into nothingness (or energy) of the matter in the universe. One prophecy of the future of Arda was that it would be remade one day, but that has yet to be seen and no prophecy is absolutely solid (usually far from it).

A lot of the Maia had beards, like the Istari, so yeah, I can see how it would be possible for an Elf to have a beard, especially in the "mortal" lands of Middle Earth. Some Maia would have beards as that is what we would expect a very wise and old individual to look like. Whether the degeneration (and hence Elves having beards) would happen in Valinor I'm not quite certain about. It is the Undying Lands and apparently ageless. It is perhaps that the Noldor (2nd clan, Tatyar, led by Finwë), the Sindar, the Teleri (3rd clan, Nelyar, led by Elwë), and others (aside from the Vanyar (Minyar, First Clan, led by Ingwë)) would all eventually age to that point, even when some of them went back to Aman to join their brethren. Being exposed to the mortal lands after the first summoning (Valian Year 4,601, around 51 VY's after the awakening of the First at Cuiviénen) seems to have the effect of aging anyone at least slightly over a long period of time. Having not seen the Light of the Two Trees also may have an impact on that. The destruction of those trees took a lot from the world and to me defines the eventual Universal "Setting of the Sun" (but, the Sun will rise again).

CÃ*rdan, who also awoke at Cuiviénen, was the oldest Elf alive in Middle Earth at the time of the War of the Ring and would be much older than Galadriel, so it's easy to see how he would be in the Third phase of life and would have a beard.

The length of the Valian Years (http://en.wikipedia.org/wiki/Valian_Years) was said to have been 9.8 solar years initially, but was increased to 144 later on. Given that, any Elf at the end of the Years of the Trees would have been 64,800 years old. With the Three Ages of the Sun up to the end of the Third Age, any Elf who was awakened at Cuiviénen would have been around 71,852 years old given the latter value of the number of solar years in a Valian Year. Given the first value of 9.8, an Elf by this time would be 11,462 years old. Quite a large difference, but keep in mind, the Elves are indeed quite immortal (in some definition of the word).

The Valar ‘fade’ and become more impotent, precisely in proportion as the shape and constitution of things becomes more defined and settled.

It could make sense in a way since the Elves live maybe up to 100,000 years (guestimated) before they fade into bodiless wraiths (lingerers) like those who stayed in Middle Earth would do after staying there throughout the Fourth Age (if they were still here now we wouldn't know it unless you were looking in the right place at the right time). The Valar probably would be around for billions of years before fading away. It kind of gives the Children of Eru a chance to sing the "Music of Creation" a second time. Look at some of the Valar, they have 'beards' in of course their corporeal forms they take in order to entertain the visual capabilities of Eru's creations.

Additional Thought on that: In theory, one may think of it this way: that in the world before the Arda that we know (another theory built upon a fictional world of course), there may have been other Children of Eru who, if faithful, would have been present in the singing of Creation. In a way, it could basically be how one becomes a Vala to replace those who faded away (reintegrated into the spirit of Eru's universe itself given what you want to believe). If one world has been created, why not one before it, and of course one after another. It's like the Rising and Waxing of the Sun and the Moon.

That last part is a far cry from Why Some Elves Have Beards, but it does get some thinking going.