Dancing on Shabbat

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 might happen to that instrument and the person might then 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 doing so knowing it is wrong.  He also wrote that some explain that nowadays it is completely permissible since we are not experts in repairing instruments and there is no concern of violating a Torah prohibition.  This helps Ashkenazic Jews, but not Sefardic Jews, as they do not rely on the Rama.  It is clear that based on the style of the Rama, the Rama was not enthralled with this leniency.  But many communities 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.  So does Shut Devar Yehoshua.    
2. In the book "Ha-Kuzari," Rabbi Yehudah Halevi, who was a Sefardic Jew, wrote that there is value in fasting and ascetic practices, but there is also value in 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 and there is thus no fear that someone will repair a musical instrument.
There is a story about Ha-Rav Moshe Feinstein that a student in his Yeshiva finally got married after many, many years.  At the Aufruf, they were so excited, including Reb Moshe, that they began to dance around the Bima.  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). 

The permission to dance therefore applies to both Ashkenazic and Sefardic Jews.  This is the reason that many communities, for many generations, have danced on Shabbat and holidays.