One of the prohibitions that apply on Tisha B’Ab is "She’elat Shalom" – greeting one’s fellow. It is forbidden according to Halacha to greet one’s friend by saying, "Shalom Alechem" or to respond to a greeting with the words "Alechem Shalom." This prohibition is codified in the Shulhan Aruch (Orah Haim 554:2).
The Halachic authorities debate the question of whether one may extend other customary greetings, such as "Good morning" and "Good evening." Hacham Ovadia Yosef rules that such greetings do not fall under the category of "She’elat Shalom" and are thus permissible on Tisha B’Ab. This is also the view of Hacham Ovadia Hadaya (1890-1969), in his work Yaskil Abdi.
It is permissible to inquire into somebody’s well being on Tisha B’Ab, such as by asking, "How are you feeling?" or "How is the fast going?"
There is a Halacha that applies all year round forbidding men from extending friendly greetings to women, which is considered immodest. It is, however, permissible for a man to wish a woman "Mazal tob" to congratulate her on her engagement, the birth of a child, or other happy occasion. On the basis of this Halacha, a number of Halachic authorities ruled that on Tisha B’Ab, too, it is permissible to wish somebody "Mazal tob," such as on the occasion of a birth or engagement.
It thus emerges that according to the strict Halacha, the prohibition of "She’elat Shalom" applies only to the greeting of "Shalom Alechem," and not to other social greetings. Nevertheless, the Poskim write that it is proper to refrain as much as possible from all social greetings on Tisha B’Ab and maintain a somber, serious demeanor as we mourn the tragedy of our Temple’s destruction.
It is forbidden to take a haircut from the Shabbat preceding Tisha B’Ab until after Tisha B’Ab. The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) rules that this applies to all the hair on one’s body, and it is thus forbidden to shave during the week of Tisha B’Ab, as well. People who must cut their hair or shave should make a point of doing so before the Shabbat preceding Tisha B’Ab.
The Ben Ish Hai adds that nail cutting is similarly forbidden from the Shabbat before Tisha B’Ab until after Tisha B’Ab. Nevertheless, if one neglected to cut his nails before the Shabbat preceding Tisha B’Ab, and they grow beyond the skin of the finger, he may cut them during the week of Tisha B’Ab. Halacha discourages allowing one’s nails to grow beyond the skin of the finger, as Tum’a (impurity) rests upon nails that grow to that point. One must always ensure that his fingernails are flush with the skin and do not extend any further. This Halacha overrides the prohibition of nail cutting during the week of Tisha B’Ab, and therefore if one’s nails grow long during that week, he should cut them in order to avoid the Tum’a that rests on long fingernails.
Summary: It is forbidden to extend the greeting of "Shalom Alechem" on Tisha B’Ab. Although other social greetings are, strictly speaking, permitted, it is proper to refrain as much as possible from social greetings in order that a somber atmosphere be maintained on this day of mourning. It is forbidden to take a haircut, shave or cut one’s fingernails from the Shabbat preceding Tisha B’Ab until after Tisha B’Ab. If one’s nails grow long and extend beyond the skin of the fingers, he may cut them even during the week of Tisha B’Ab.
