The Shulhan Aruch (Orah Haim 339:3) writes that the Sages enacted a prohibition against dancing on Shabbat, out of concern that one might repair an instrument in order to provide musical accompaniment. The Rama (Rav Moshe Isserles of Cracow, 1530-1572), commenting on this Halacha, observed that many Ashkenazic communities were lenient in this regard, and allowed dancing on Shabbat without the Rabbis protesting. To explain this phenomenon, the Rama first writes, "Mutab Yiheyu Shogegin Ve’al Yiheyu Mezidin" – the Rabbis realized that their protests would be ignored, so they preferred that the people violate this prohibition inadvertently, unaware that Halacha forbids dancing on Shabbat. According to this explanation, the Rama concedes that dancing on Shabbat is prohibited, and those aware of this Halacha should refrain from dancing. But the Rama then adds a second possibility, suggesting that since in modern times most people do not know how to make or repair music instruments, and this is done only by professional artisans, this prohibition is no longer applicable. According to this theory, it is permissible to dance on Shabbat.
Hacham Ovadia Yosef writes that when the Rama presents two different views in this format, he sides with the first opinion. Therefore, the Rama’s view is that dancing is forbidden on Shabbat. Accordingly, Hacham Ovadia writes that even Ashkenazim, who follow the Rama’s rulings, should refrain from dancing from Shabbat.
"Dancing" for the purposes of this Halacha refers to a dance in which one jumps off the ground. If people walk around in a circle, this is not considered "dancing" and it is thus permitted on Shabbat.
An exception to this prohibition is Simhat Torah, when dancing is allowed for the sake of rejoicing with the Torah. However, even on Simhat Torah, one should not shake the Rimonim (the decorative metal pomegranates that adorn a Sefer Torah) to make music.
Some Halachic authorities felt that it is permissible to bring non-Jews to play music on Simhat Torah, but common custom does not follow this opinion.
The Shulhan Aruch then proceeds to note several prohibitions enacted by the Sages out of concern that these actions might lead one to write on Shabbat. For example, a Bet Din does not convene on Shabbat, even though all they do is listen to arguments and study and discuss the relevant source material. Another example is Kiddushin – the first stage of the marriage process, when the groom gives the bride a ring. The Hupa, the second stage of the process, is certainly forbidden, as it constitutes a "Kinyan" (legal transaction) in that the mutual responsibilities of a husband and wife take effect at the time of the Hupa. The prohibition against making weddings on Shabbat applies as well to "Yibum" and "Halisa."
The Shulhan Aruch also writes that when one marries a widow before Shabbat, the couple should not consummate the marriage on Friday night, as in such a case, the consummation of the marriage effectuates the "Kinyan" of marriage. Of course, this is not so relevant nowadays, when weddings are rarely held on Friday.
One may not perform a Pidyon Ha’ben on Shabbat, even if one wishes to give a Kohen an object of value – such as a silver Kiddush cup – instead of money. Even though the object is not Mukseh, the Pidyon should not be performed on Shabbat. Likewise, one may not give the Kohen the money for the Pidyon Ha’ben before Shabbat and stipulate that the transaction should take effect on Shabbat.
Divorces are not done on Shabbat.
Summary: It is forbidden to dance – defined as jumping off the ground – on Shabbat or Yom Tob, though on Simhat Torah this is allowed. A Bet Din may not convene on Shabbat, and neither stage of a wedding – neither the Kiddushin nor the Hupa – may be held on Shabbat. A Pidyon Ha’ben may not be performed on Shabbat, and divorces are not done on Shabbat.