Beard

A beard was a form of hair growth on one's face. They were often shortened to just thick upper lip hair known as a moustache.

The Master
The Master, in multiple incarnations, (TV: Terror of the Autons, The Keeper of Traken) often wore one known as a goatee. (COMIC: The Judas Goatee)

The incarnation of the Master that caught up with the First Doctor sported a beard, (AUDIO: The Destination Wars) as did the Master that frequently fought the Third Doctor and UNIT. (TV: Terror of the Autons-Frontier in Space)

Tremas, a Trakenite, sported a long grey beard when he was killed as his body was possessed by the Master. During this process, his hair shortened and changed to brown. The Master retained this beard, (TV: The Keeper of Traken) which led to the Fifth Doctor asking the Tenth if he "still had that rubbish beard". The Tenth Doctor replied that he had a wife instead. (TV: Time Crash)

The Tzun incarnation wore a Van Dyke beard. (PROSE: Housewarming)

Following his regeneration, (COMIC: Fast Asleep) the white-haired incarnation of the Master that emerged during the Last Great Time War grew a scruffy, white goatee which he wore up to the point he transformed himself into an infant human, (AUDIO: Beneath the Viscoid, The Heavenly Paradigm) who would be named "Yana". Unlike his Time Lord self, Yana remained clean-shaven after he matured, and it was in this form that the Master returned before regenerating into a brown-haired incarnation, (TV: Utopia) who would adopt the alias of "Harold Saxon". (TV: The Sound of Drums) This Saxon Master recalled two earlier incarnations whom he called and  respectively. (PROSE: The Secret Diary of the Master)

When the Twelfth Doctor reencountered the Master in his "Harold Saxon" incarnation, he had grown a goatee. (TV: World Enough and Time)

The Spy incarnation of the Master who confronted the Thirteenth Doctor sported a beard, (TV: Spyfall) which grew bigger as he posed as Rasputin during his master plan. (TV: The Power of the Doctor)

The Doctor
The Doctor first grew a beard in his fifth incarnation when he spent over a year stuck in Victorian London, as people thought he was too young to take seriously without it, (AUDIO: The Haunting of Thomas Brewster) but he apparently shaved it off once he resumed his travels. He grew a beard once again when he was aged a relative year while travelling through a time distortion field to escape a high-security prison. (AUDIO: Doing Time)

While staying in the house on Henrietta Street in his eighth incarnation, the Doctor briefly grew a neat goatee beard, intended to test if he could change as well as the wider universe; he claimed to have worked out how to transform himself into a creature of pure light by adapting his self-biology, but estimated that it would take three thousand years of concentration, so growing a beard was easier. (PROSE: The Adventuress of Henrietta Street) He only kept the beard for a short time before he shaved it off. (PROSE: Mad Dogs and Englishmen, Hope) He eventually grew a stubble which he had right up to his death. (TV: The Night of the Doctor)

The incarnation known as the War Doctor grew a stubble when younger. (COMIC: Ambush) As he grew older, he grew a brown Van Dyke beard (COMIC: The Lost Dimension) which became grey with age, (AUDIO: The Innocent; COMIC: The Whole Thing's Bananas; TV: The Name of the Doctor, The Day of the Doctor) leading Kalan to nickname him "Greybeard". (AUDIO: The Eternity Cage)

While trapped for five days on Hurala, the Tenth Doctor was able to prevent himself from growing a beard through sheer concentration. When his rescuers found him, he appeared clean-shaven, but noted he would soon experience a terrible itch. (PROSE: Prisoner of the Daleks)

The Eleventh Doctor would grow a brown beard when held in captivity for significant periods of time. (TV: Day of the Moon, The Wedding of River Song)

An alternate Twelfth Doctor wore one, too. (COMIC: Four Doctors)

The Fourteenth Doctor grew a stubble. (TV: The Power of the Doctor)

Others
The War Chief had a white beard in his deformed incarnation. (PROSE: Timewyrm: Exodus)

In the primary universe, Professor Eric Stahlman, the founder and head scientist of the Inferno Project, had a beard, whereas his counterpart in the Inferno Earth universe, Eric Stahlmann, was clean-shaven and also wore dark-tinted glasses. (TV: Inferno)

Charles Dickens had one when he met the Ninth Doctor and Rose. (TV: The Unquiet Dead)

Alistair Gordon Lethbridge-Stewart sported a white beard in his later life. (TV: Enemy of the Bane)

A parallel Davros had a white beard when he met the Eighth Doctor and Bliss before the Dalek Time Strategist's tampering caused him to gain the appearance and personality of his N-Space counterpart. (AUDIO: Palindrome)

Some Sontarans also had vestigial hair, generally in the form of beards, which grew white with age. (TV: The Two Doctors; PROSE: Shakedown; COMIC: The Betrothal of Sontar, Supremacy of the Cybermen)

A version of the Corsair who wore a black top hat had a dark beard. (PROSE: Eleven Things You Probably Didn't Know About the Corsair)

In a Kantrofarri-induced shared dream, Santa Claus had a beard when he met the Twelfth Doctor. According to Wolf, he initially grew the beard as a disguise, but "people have picked up on it". (TV: Last Christmas) Indeed, even the Roboforms, who took the form of robot Santas, had beards. (TV: The Christmas Invasion, The Runaway Bride)

Owen Harper likened his partial resurrection to Jesus Christ "but without the beard". He immediately realised "shit, I'm never gonna have a beard" but denied wanting one. (TV: A Day in the Death)

Varun Singh had one grown due to the practice of Kesh in Sikhism. (TV: For Tonight We Might Die, The Coach with the Dragon Tattoo, Brave-ish Heart)

Abby McPhail met Cleo Proctor and Shawna Thompson at university and bonded over their mutual love of murder podcasts and hatred of their "alt-right neckbeard neighbour". (AUDIO: SOS)

Behind the scenes

 * FASA's Doctor Who Roleplaying Game depicted Davros as having a goatee before his fateful accident, after which he appeared hairless.