According to the Shulhan Aruch (Orah Haim 493:2), the custom is that haircuts and shaving are prohibited during the Omer, up until the 33rd or 34th days (see earlier Daily Halacha regarding on which day these become permitted). The Gemara (Yevamot 62b) describes this period as a time of mourning since Rabbi Akiva’s students died.
Rabbi Itzhak Ben Oualid (Vayomer Itzhak, Orah Haim, Likute Dinim, §§ 2), quoting Rabbi Yishmael HaKohen (Zera Emet), says that if one needed to meet with a king it was customary to present oneself in a well-groomed manner. Since being unkempt would be considered below the dignity of a king or any pre-eminent person, it was the custom to be lenient and permit haircuts in such a situation. It follows that if there is no such custom in a certain locale or if that there is no pressing need to get a haircut, then it would not be permitted. Nevertheless, Rabbi Shlomo Auerbach (Ma’adane Shlomo) and Rabbi Moshe Feinstein (Igrot Moshe, Orah Haim, vo. IV, § 102) write that if one were to suffer a financial loss as a result of not being groomed, one would be permitted to shave. Rabbi Yosef Shalom Elyashiv (Hilchot Hag BeHag, Sefirat HaOmer, ch. 7, note 65), on the other hand, says that even in the case of financial loss, one is should avoid shaving during this period. If one is faced with a situation where it is vital to shave, then it is preferable to shave on Friday. This is based on Rabbi Yehuda Shmuel Ashkenazi (Bet Oved) who writes that there is a custom to shave on Fridays during the Omer in honor of Shabbat, and indeed this is the custom of the Yemenite community. According to Rabbi Yehoshua Maman (Emek Yehoshua, vol. VI, ch. 33) the Moroccan custom with respect to shaving every Friday is to be strict and not permit it. Furthermore, when Lag Ba’Omer falls on a Sunday, those who follow the custom of taking haircuts on the 33rd day, may have one’s hair cut on the preceding Friday, as will occur this year (5777/2017). The Rama (ibid.) and Rabbi Yosef Benaim (Noheg Behochma, p.49) state that the honor of Shabbat takes precedence over the mourning of the Omer, specifically in anticipation of Lag BaOmer.
Regarding women, Rabbi Ben Zion Abba Shaul (Or LeZion, vol. III, ch. 17, § 3) says that the stringency of mourning of the Omer period is comparable to that of the Shloshim period, the thirty days after a relative passes away (h’v). Just as women are not forbidden from getting a haircut during the Shloshim, they are not forbidden from doing so during the Omer. The same is true for trimming one’s eyebrows and other types of grooming.
Additionally, Rabbi Itzhak ibn Denan (LeItzhak Re’ah, Likute Orah Haim) writes that the Moroccan custom is to permit cutting one’s nails, whether an man or a woman, during the Omer.
Summary: Under extenuating circumstances, one may shave during the Omer, but should preferably do so on Friday. Women may have haircuts during the Omer. Nail-cutting is permitted during the Omer. If Lag BaOmer falls on Sunday, like this year (5777/2017), those who have the custom to shave on the 33rd day may do so on the preceding Friday.