I wanted to talk about individual white hairs in a beard. I was taught that individual hairs where always 毛。So I said:
一些白毛已经出现了
This was immediately corrected to:
一些白发已经出现了
Shall I discard the rule that I learned from my professor? Is there a better rule, which I can apply in its stead? Should 毛 perhaps be used only with the measure word 根？
Generally speaking, hair that grows out of the top of your head is 髮(发), hair around a man's mouth is 鬍鬚(胡须), and hair in other parts of your body is 毛. 毛 is usually specified by the body part, e.g.
鼻毛 nostril hair，
腋毛 armpit hair，
睫毛 eye lash
胸毛 chest hair
眉毛 eye-brow hair
Note that 眉毛 can refer to individual eye-brow hair, or the whole eye-brow itself.
頭髮 is often used instead of just 髮
If you want to talk about individual white hair in your beard, it is 白鬍鬚（白胡须）。白发 or 白頭髮 is white hair on your head.
根 is the measure word for individual strands of hair.
抱歉，您的表达有点问题。我无法理解你 “具体”想要表达什么。 对于胡子上面长的毛，我们一般称为“胡须”。 你可以这样使用： 我的一些胡须已经变白了。
量词，如果是很多，比如 四个以上，我建议 您 使用 “一些”，“有些”等量词。
祝您生活愉快！
The correction 一些白发已经出现了 is worse than what you were saying 一些白毛已经出现了, with references to the beard, or mustache. Although 毛 is rarely used to describe the hair around your lips or jaws, it is not completely out of line, but the use of 发 is.
The famous use of 毛 for such case is the old saying: "嘴上無毛,做事不牢". Another saying that refers to 毛 as hair is "毛頭小子. The former refers to the unreliability of a young person. The latter refers to the immaturity and inexperience of the young person.
However, while 毛 can mean the hair on head, such as 他頭上僅存的幾根毛..., 发 can't be used to describe any body hairs but on the head.
You can modify your sentence to "我嘴上長出了一些(幾根)白毛". With referring to your mouth, I think the sentence is acceptable, although "我嘴上長出了一些(幾根)白鬍子(鬚)" is more proper and formal.
对于人体除了头发以外的毛发，中文一般“部位+毛”的形式来表达。不过，胡须是一个例外，我们一般就称之为胡子或者胡须，不用“毛”或者“发”。比如你所描述的情况通常就应该表达为“我长了一些白胡子”。 不过有时候我们也会用不加部位的“毛”这个字来开玩笑，比如你可以说你自己胡子里长了一些白毛，或者说你红头发的密友“长了一头红毛”。但是一般不用“发”来形容头发以外的毛发，开玩笑也不会用。
