Question: Is it permissible to dance on Shabbat?
Answer: The Mishnah and the Gemara in Beitzah (36b) say that it is forbidden to dance on Shabbat and holidays out of a concern that someone might play an instrument, something then might happen to the instrument, and the musician might repair it, which is a Torah prohibition. Today, however, it is permissible to dance, for three reasons:
1. The Rama in the Shulchan Aruch (Orach Chaim 339:3) wrote that we do not protest dancing on Shabbat, since people are already accustomed to this activity, and it is better for them to perform it unwittingly than to do so knowing that it is wrong. He also wrote that some explain that nowadays it is completely permissible to dance because we are not experts in repairing instruments and there is no concern that we will violate this Torah Mitzvah. This only helps Ashkenazic Jews as Sefardic Jews do not rely on the Rama. Based on the style of the Rama, it is clear that he was not enthralled with this leniency, but many communities nevertheless do dance on Shabbat, and not only Religious-Zionists. In Shut Minchat Eleazar (1:29), Ha-Admor of Munkatch, who was definitely not a Religious-Zionist, wrote at length that it is certainly permissible to dance, as does Shut Devar Yehoshua (4:42). They both permit dancing as is the custom of many communities.
2. In the book "Ha-Kuzari," Rabbi Yehudah Halevi, who was a Sefardic Jew, wrote that there is value to fasting and ascetic practices, but there is also value to rejoicing, and our dancing on Shabbat and holidays is no less of Divine worship than fasting and ascetic practices. This means that there is a Sefardic Rishon (Rabbi of the Middle Ages) who permitted this activity.
3. The Aruch Hashulchan (ibid. #9) wrote that the concern and the reason for the prohibition are only when people dance to a precise rhythm, but what people do today is not considered "dancing." People go around in a circle and jump up and down. People do not dance in a way that it must be accompanied by musical instruments. There is thus no fear that someone will repair a musical instrument as a result of the dancing. There is a story about Ha-Rav Moshe Feinstein: a student in his yeshiva finally got married after many, many years. At the Aufruf, they were so excited that they began to dance around the Bima. Ha-Rav Feinstein participated. A student asked him: Isn't it forbidden to dance on Shabbat? Ha-Rav Feinstein responded: You call this dancing?!
And Ha-Rav Chaim Kanievski related that he once asked the Chazon Ish about dancing on Shabbat for an Aufruf or Bar Mitzvah, and the Chazon Ish answered that the custom is to be lenient. He said, however, that his father, the Steipler, would walk around and not dance (Ma'aseh Ish vol. 5 p. 17).
Permission to dance therefore applies to both Ashkenazic and Sefardic Jews. This is the reason that many communities have, for many generations danced on Shabbat and holidays.