At least, that's what I was taught as a boy. Then I arrived in Israel for my year abroad, and I discovered that our rosh yeshiva, Rabbi Dr. Aharon Lichtenstein (who was already remarkable for being generally clean-shaven) would shave during Sefira in honor of Shabbat, as well as during the last weeks of Tammuz. His logic was based on the approach of his father-in-law, Rabbi Dr. Joseph B. Soloveitchik, the champion of Modern Orthodoxy in America.
The argument goes like this: the national mourning periods of Sefira and the Three Weeks are modeled after personal mourning. Personal mourning has three levels: the week of shiva, the month of sheloshim and the year of yud-bet hodesh. Similarly, the period of national mourning we are currently in has three levels, going in reverse: the Three Weeks, the Nine Days and then the Fast of 9 Av itself. At the lowest level, which is common to both the Three Weeks and Sefira, one may shave when one's stubble becomes socially unacceptable (presumably never in Brooklyn). Those who follow this view are very busy right now, since this evening we transition from the Three Weeks to the Nine Days. (You can read more here.)
There is beauty to this mechanistic approach. The halakhic mind craves order, and this method imposes order upon that most chaotic and unpredictable element of Jewish life, custom. We just plug in the formula, and it all works smoothly.
However, as I began my own studies for the rabbinate, I had to confront the sources myself, and things stopped being so simple. You see, the Mishna (Taanit 4:7) does not talk about restrictions of the Three Weeks; it merely says, "During the week in which 9 Av falls, one may not cut hair or launder." This, in fact, is what R. Joseph Karo writes in Shulhan Arukh (OH 551:3), and this is the Sephardic custom. As R. Karo stresses (12): "During this week, haircutting--whether of the head or any other hair--is forbidden." It is the Ashkenazic authority R. Moses Isserles who glosses (4): "With haircutting, the custom is to be stringent from 17 Tammuz." Are we supposed to assume that only the Ashkenazic custom uses the analogue of personal mourning? The implication seems clear: what you do with your shaver during the Three Weeks is all about your community of origin.
This led me to an interesting dichotomy: during Sefira, shaving should be allowed, while during the Three Weeks, it should be disallowed. This is a conclusion both more lenient and more stringent than that of R. Soloveitchik and R. Lichtenstein. It was and remains an uncomfortable position, conforming neither to common practice nor to the view of my rebbe.
Now, many may see this as the search for niggling minutiae, arcane details of tribal customs from centuries past. I disagree. As attractive as it may be to approach Judaism as programmatic and predictable, I see it as an organism, wild and spontaneous. It is a celebration of tradition and transcendence, revelation and revolution, and I cannot wait to see what happens next.
At least it keeps my mind off this scratchy beard.