During the period of Sefirat Ha'omer, we refrain from certain forms of festivity as an expression of mourning for the tragic deaths of Rabbi Akiva 24,000 disciples, which occurred during these weeks. The Shulhan Aruch mentions that we do not conduct weddings, or cut our hair during this period. The Magen Avraham (commentary to the Shulhan Aruch by Rabbi Avraham Gombiner, Poland, 1637-1683) further adds the prohibition of listening to music.
The question was raised as to whether we must also refrain from other activities of a festive nature. For example, during the period of Ben Ha'mesarim – the three weeks between Shiva Asar Be'Tamuz and Tisha B'Av – Halacha forbids reciting the joyous Beracha of "She'he'hiyanu." This Beracha is recited on joyous occasions – such as upon partaking of a new fruit or wearing a new garment – whereby it expresses gratitude to God for bringing us to the given occasion. This expression is incompatible with the Ben Ha'mesarim period, during which numerous calamities befell the Jewish people. Should we extend this line of reasoning to the Sefira period, as well, which also marks a time of great tragedy, and forbid reciting "She'he'hiyanu" during these seven weeks?
The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909), in his work Or Ha'haim, draws a fundamental distinction in this regard between the periods of Ben Ha'mesarim and Sefirat Ha'omer. As we have unfortunately seen throughout Jewish history, the period of Ben Ha'mesarim is intrinsically designated as a time of misfortune; it is inherently defined as a time of tragedy, and it would therefore be inappropriate to recite the joyous Beracha of "She'he'hiyanu" during this period. Sefirat Ha'omer, by contrast, is actually a very auspicious time. So much so, that the Ramban, in his Torah commentary (Parashat Emor), speaks of these weeks as a kind of "Hol Ha'mo'ed" in between the two festivals of Pesah and Shavuot. The Zohar describes the Sefira period in these terms, as well. Although this period saw the tragic death of Rabbi Akiva's students, this calamity does not characterize these weeks as a time designated for misfortune. Therefore, although we indeed refrain from certain forms of festivity, we need not go beyond the areas namely, weddings, haircuts and music. All other festive activities are permissible, even those which are forbidden during Ben Ha'mesarim, such as reciting "She'he'hiyanu." This is the view taken by several other authorities, as well, including the Yafeh La'lev and Pahad Yishak. Hacham Ben Sion Abba Shaul (Israel, 1924-1998) likewise follows this position, in his work Or Le'sion (vol. 3).
Similarly, Hacham Ben Sion rules that it is permissible to purchase a new garment during the Sefira period, despite the joy it brings a person. By the same token, Hacham Ovadia Yosef (Hazon Ovadia – Laws of Yom Tov, p. 74) rules that one may move into a new home during Sefira, or renovate his home. This includes expanding, painting and refurnishing. One may also host a "Hanukat Ha'bayit" celebration during Sefira in honor of his moving into a new home, just as engagement parties may be held during the Sefira period, provided that no music is played.
Summary: During the period of Sefira we refrain from making weddings, listening to music, and haircutting. One may, however, purchase a new garment, recite the Beracha of "She'he'hiyanu," move into a new home, renovate one's current home, and host a celebration (without music) in honor of an engagement or a new home.