May a man shave his beard in a kosher manner on Chol Hamoed?
(Sefira considerations aside)
Short Version: Rav Yosef Dov Soloveitchik and others are of the opinion that it is permitted for one who regularly shaves to shave, and since it is permitted it is a mitzva to do so, so as not to look disgraceful on chol hamoed (Source). Rav Moshe Feinstein makes a similar argument in Igrot Moshe OC I 163.
Long Version: The Mishna in Moed Katan 3:1 (page 13b in the Bavli) lists 7 people who are allowed to get haircuts on chol hamoed, implying that everyone else is forbidden to get a haircut on chol hamoed. The gemara there (14a) wonders why is this so? Even though cutting hair is one of the 39 categories of forbidden labor ("Gozeiz"), shouldn't it be permitted on chol hamoed by the rule of "tzorech hamoed" -- that labor which is performed for the sake of the holiday may be performed on chol hamoed? Surely getting a haircut to look nice is for the sake of the holiday! The gemara answers that there is a rabbinic decree forbidding haircuts on chol hamoed, to encourage people to get their haircuts before the holiday and not wait until they are free from work on chol hamoed. The mishna's exceptions are cases of people who could not for various reasons get a haircut before the holiday; for them there is a dispensation to allow haircuts on chol hamoed because the end goal is to have people look nice during the holiday.
Now comes the important question: can we extend beyond the seven cases of the mishna? The gemara asks about one who was busy all of Erev Chag searching for a lost object and did not have an opportunity to get a haircut before the holiday. The gemara answers that he may not get a haircut on chol hamoed as his having been stuck is not obvious and eveyone may come to claim that they were stuck and can take a haircut. Rather, it says, only someone whose reason for not getting a haircut is apparent to all is permitted to get a haircut on chol hamoed.
At this point it should be clear that a man who is regularly clean shaven may shave on chol hamoed, for it is obvious to all that he was unable prior to the holiday to shave the hairs that hadn't grown yet! And since it is permitted for him to shave, it becomes a mitzva for him to do so, because as we said, the end goal is to have people look nice during the holiday.