Gemara

(Beraisa): An animal may go out in a Chatzer with a bell if the clapper is plugged up.

148b (and Beitzah 30a) (Mishnah): On Yom Tov we may not clap hands on each other or over the heart, or dance.

Question (Rava bar Rav Chanan): People do this, and we do not protest!

Counter-question (Abaye): Rava taught that one should not sit even with the Lechi (a post in an alley). People sit there, and we do not protest!

Answer to both questions (Abaye): We are silent because it is better that Yisraelim transgress b'Shogeg than b'Mezid!

Yoma 19b (Mishnah): If the Kohen Gadol was about to fall asleep (on Yom Kipur night), young Kohanim snapped their fingers in front of him.

Rishonim

Rif and Rosh (Beitzah 20a and 5:2): The Yerushalmi forbids clapping out of anger, like Balak, and willingly. It says that in the days of R. Yonah, Chachamim clapped on the back of the hand. They did so at the wedding of R. Shimon bar Rebbi. R. Meir passed by and heard this, and said 'Rabanan permitted Shabbos!' Rebbi heard (unaware of who said this) and asked 'who came to be an authority over us in our house?!' When R. Meir heard Rebbi's voice, he fled.

Rambam (Hilchos Shabbos 23:4): One who fixes a Kli in any way is liable. Therefore it is forbidden to make a musical sound on Shabbos, whether with a Kli Shir (musical instrument) like a harp or lyre, or with other things. One may not even bang a finger on the ground or a tablet, or snap fingers the way singers do, or rattle a nut for a child, or play with a bell to quiet a child. We decree against all this and similar things. This is a decree lest one fix a musical instrument.

Kesef Mishneh: They snapped fingers in front of the Kohen Gadol, because Ein Shevus b'Mikdash. (Needs of the Avodah in the Mikdash override Isurim mid'Rabanan.)

Rambam (5): (On Shabbos) we may not clap hands on each other or on the thigh, or dance. This is a decree lest one fix a musical instrument. One may clap unskillfully.

Tosfos (Beitzah 30a DH Tanan): Rashi explains that we forbid clapping and dancing, lest one fix a Kli Shir. However, it is permitted for us. In those days, they were expert at making Klei Shir, so it was reasonable to decree. We are not experts at making Klei Shir, so it is not reasonable to decree.

Beis Yosef (DH v'Lo): In Beitzah (5a) we say that even though the reason for the decree is Batel, the decree is not Batel. However, Tosfos (Avodah Zarah 35a DH Chada) compares this to exposed liquids, which are permitted nowadays that snakes are not common.

Maharik (9): Our custom is to dance on Simchas Torah, even several old Chachamim, when we sing praises of the Torah. This (dancing on Yom Tov) is Shevus. We permit it for the honor of Torah. Rav Hai Gaon said so.

Beis Yosef (DH v'Chasav): Maharik brought from Rav Hai Gaon that dancing on Simchas Torah to honor the Torah overrides Shevus. They did not distinguish between the days of the Gemara and nowadays, unlike Tosfos. Also the Poskim wrote Stam, and did not distinguish.

Terumas ha'Deshen (62): One may snap fingers to quiet a child. Even though a Mishnah forbids clapping, we learn from the Kohen Gadol. If he was about to doze, young Kohanim snapped fingers in front of him to prevent sleep (Yoma 19b). Do not say that this is because Ein Shevus b'Mikdash. If so, the Gemara should have said that this is why it is permitted, like it says about putting a metal coal in the water (to blunt the chill for the Kohen Gadol, who immerses in it), even though this solidifies the metal. Indeed, it is not in Rashi's text, but it is in the text of Tosfos and the Rosh. We prove from them that the Heter is not due to Ein Shevus b'Mikdash.

Question (Beis Yosef DH umeha'Rav): Why did the Terumas ha'Deshen permit, unlike the Rambam?

Poskim

Shulchan Aruch (OC 339:3): (On Shabbos) we may not clap hands on each other or on the thigh, or dance. This is a decree lest one fix a musical instrument.

Mishnah Berurah (7): This is forbidden whether amidst mourning or anger, to arouse pain, or due to Simchah. It is all a decree lest one fix a Kli Shir to arouse mourning or Simchah.

Mishnah Berurah (8): It is permitted on Simchas Torah due to Kevod ha'Torah, since the Isur is only Shevus. We do not permit for other Simchah of Mitzvah, e.g. a Nesu'in. Even to honor the Torah, one should permit only clapping and dancing, but not clanging bells, and surely not other Klei Shir.

Kaf ha'Chayim (10): One may not dance in front of a Kalah on Shabbos.

Shulchan Aruch (ibid.): One may not even bang a finger on the ground or a tablet, or snap fingers the way singers do.

Magen Avraham (2): Even though they used to snap fingers in front of the Kohen Gadol, that was in order to wake him. It was not a musical sound. This is why the Shulchan Aruch says "like singers."

Eshel Avraham (2): Rashi (Yoma 19b DH Tzarasah) says that they snapped abnormally. Tosefes Shabbos derives from this [that one may not snap normally on Yom Tov even not in a musical way]. He is correct.

Kaf ha'Chayim (12): The Magen Avraham holds that the Heter was not due to Ein Shevus b'Mikdash. Therefore he needed to say that it was not a musical sound.

Mishnah Berurah (9): The Magen Avraham permits snapping loudly in order to wake someone. The way Tosefes Shabbos explains Rashi, it seems that Rashi forbids also this. The Eshel Avraham connotes like this.

Bi'ur Halachah (DH Oh): Why did the Terumas ha'Deshen write unlike the Rambam? Our text in the Rambam is "on a tablet one Keneged (opposite, or corresponding to) one." It refers to what it says above, he hit on the ground or on a tablet, one Keneged one. This is like singers do, but not if he hit just once on the ground or tablet. He hints that one may rap on a door or similar matters, like he says in Perush ha'Mishnayos at the end of Eruvin. We can say that snapping fingers is not Derech Shir (the way of singing). Perush ha'Mishnayos in Yoma says that they made nice sounds. We need not say that this is Derech Shir. Perhaps he just explains that they did not bang on other things, for this is less honorable in front of a Kohen Gadol than snapping fingers.

Shulchan Aruch (ibid.): One may not rattle a nut for a child, or play with a bell to quiet a child. We decree against all this and similar things, lest one fix a musical instrument.

Gra (DH Oh): We learn from 53a. (It permits a bell on animal only if it is plugged up.) Also the Yerushalmi says that [a slave] may go out with a bell, as long as he will not cause it to ring.

Shulchan Aruch (ibid.): One may clap k'Le'acher Yad. (Normally, this means k'Le'acher Yad. It seems that here it means on the back of the hand.)

Beis Yosef (DH Kasav): The Rambam permits clapping k'Le'acher Yad. He learns from the Yerushalmi. Even though it connotes that R. Meir forbids, the Chachamim in the days of R. Yonah are Basra. They hold like Rebbi's household. Also, perhaps even R. Meir permits. He thought that they were clapping normally. When he heard Rebbi's response, he realized that they were clapping on the back, so he fled out of shame that he improperly rebuked them. Semag says that the Yerushalmi connotes that Rebbi permits this. We may rely on him, since Rabanan were frequently by Rebbi, so his teachings were very precise.

Bi'ur Halachah (DH u'Lesapek): The Yerushalmi discusses clapping k'Le'acher Yad. This implies that for clanging a bell or other things that are more Derech Shir, it is forbidden even k'Le'acher Yad. This is proper, for even for clapping there is not a clear Heter from the Yerushalmi, since R. Meir was stringent. (The Beis Yosef's answer is difficult.) It is hard for us to permit more than the Yerushalmi explicitly permits.

Rema: Nowadays, people clap and dance and we do not protest, for it is better that Yisraelim transgress b'Shogeg than b'Mezid. Some say that nowadays everything is permitted, for we are not experts about making Klei Shir, so we do not decree about fixing one, for it is not common. Perhaps this is why people are lenient about all this.

Mishnah Berurah (10): We are lenient about clapping and dancing, but not about the other matters in this Sa'if. Even for clapping and dancing, one should not be lenient if not for a Mitzvah. We merely say that it is better that Yisraelim transgress b'Shogeg than b'Mezid.

Kaf ha'Chayim (13): Maharshal says that even though the Poskim did not distinguish nowadays from the days of the Gemara, if it is for a Mitzvah, we may be lenient. However, the Acharonim are stringent even for the Mitzvah of Nisu'in.

Kaf ha'Chayim (14): One who fears Shamayim should be stringent for himself and for his household.