After Yom HaAtzmaut was established as a day of joy and thanksgiving, the question arose: do the customs of mourning that we observe during the Sefirat HaOmer period apply to Yom HaAtzmaut? The halachic authorities have expressed their opinions in both directions. The accepted practice is not to keep customs of mourning that diminish expressions of the joy. Therefore, it is permissible to dance and play music. However, one should not make a wedding on that day, because avoiding weddings is not considered an expression of mourning that clashes with the joy of Yom HaAtzmaut.
Those who shave regularly should shave in advance of Yom HaAtzmaut, just like one puts on special clothing before the holiday begins. Regarding haircuts, it seems that only one who looks disgraceful because of his [long] hair may take a haircut prior to Yom HaAtzmaut. Someone who looks fine, however, is allowed to take a haircut only on Yom HaAtzmaut itself, for then the joy of the day overrides this custom of mourning12.
The Chief Rabbinate, under the leadership of Rabbi Unterman and Rabbi Nissim, determined that even Ashkenazim who observe the customs of mourning during the latter part of Sefirah should not curtail their joy on the 28th of Iyar, Yom Yerushalayim. After all, many communities terminate all the restrictions after Lag BaOmer (the 33rd day of the Omer) (see above 3.2-4). This is all the more so, now that the 28th of Iyar has been instituted as a day of thanksgiving and joy over the miracle that HaKadosh Baruch Hu performed for His nation, Israel. Therefore, one may even make a wedding on Yom Yerushalayim.
We do not say Tachanun on Yom HaAtzmaut, Yom Yerushalayim, or during the Minchah service preceding these days (Peninei Halachah, Tefillah 21.7; see also 21.2, n. 1).
 Rishon Letzion, Sefardic Chief Rabbi Yitzchak Nissim, writes (Hilchot Yom HaAtzmaut VeYom Yerushalayim, pp. 334-40) that all customs of mourning are canceled on Yom HaAtzma’ut. He bases his ruling on those poskim who hold that one who has not yet fulfilled the mitzvah of procreation may get married during Sefirah (Radvaz, Pri Chadash). Similarly, some people take haircuts in honor of Shabbat and Rosh Chodesh (Radvaz, Ya’avetz). Furthermore, Rabbi Chayim Palagi writes (Mo’ed LeChol Chai 6) that a miracle happened to some of the people of his city on the eighth of Iyar, and to others on the eleventh, and they take haircuts on these days. These are Rabbi Nissim’s proofs. There is even more room for leniency when it comes to shaving, because shaving is not festive in nature; it simply eliminates the mournful appearance, as we explained above (3.7). The author of Responsa Yaskil Avdi (6:10), on the other hand, does not permit haircuts or weddings on Yom HaAtzmaut. Our master and teacher, HaRav Tzvi Yehuda HaKohen Kook, used to rebuke those students who usually shave but look like mourners on Yom HaAtzmaut, saying, “Their countenance testifies against them (cf. Yeshayah 3:9) that they are not happy and that they do not truly thank Hashem for the miracle.” See Rabbi Shmuel Katz’s essay in HaRabbanut HaRashit, vol. 2, pp. 877-82.
Regarding mourners, Rabbi Katz’s abovementioned book (p. 900, n. 37) cites a responsa from Rabbi Goren, ruling that, similar to Chanukah, the holidays of Yom HaAtmaut and Yom Yerushalayim do not cancel shiv’ah (the seven day period of mourning for a close relative). Therefore, a mourner does not recite Hallel, nor is Hallel said by others in a house of mourning, rather elsewhere. If the mourner has completed shiv’ah, he should join the festive prayers and celebrations, as long as there is no live music. A mourner may not take a haircut during shloshim (the thirty day period of mourning for a close relative) in honor of these holidays.