Rhythm and Dancing on the Day of Shabbos
Based on this article of Rabbi Avi Weinrib
The Mishna in Beitza [36b] rules that during Yom Tov, it is forbidden to clap hands, or slap one's thighs or dance.
Tosafos [Shabbos 148b] explain that such actions generally accompanied playing of musical instruments, which may require adjustment or repair, both of which are forbidden on Yom Tov or Shabbos, either as the Melacha D'Oraisa of Maka BiPatish or the Rabbinic prohibition of Tikun Maneh.
Most Poskim agree that this applies today as well, even though Tosafos [Beitzah 30a] maintain that it is no longer relevant. Tosafos is actually quite remarkable; firstly, they seem to ignore the guideline that a decree can only be revoked by a Beis Din greater both in numbers and wisdom. [See Bais Meir 339-1 and Igros Moshe OCh. 2:100]
Secondly, Tosafos appear to be permitting, now that the decree is no longer relevant, the playing of musical instruments. However, most Poskim explain that Tosafos only permit clapping and dancing, not the playing of actual instruments. [Shut ShaAr Ephraim 36; Eliyahu Raba OCh 339:1; Biur Halacha ibid s.v. ulisspek]
The Shulchan Aruch [OCh 339:3] rules that it is forbidden to clap hands, slap one's thighs or dance on Shabbos. This decree includes drumming on a table or the like with or without any utensil, and also snapping one's fingers.
The Mishna Berura [S.K. 8] only permits this on Simchas Torah where clapping and dancing is a Mitzvah in honor of the Torah. However. For any other reason, even other Mitzvos such as a Sheva Berachos it would not be permissible.
Chasidim tend to be more lenient based on the Minchas Elozer [Volume 1 Siman 29] who permits dancing and singing for those who are caught up in the joy of Shabbos as it is considered a Mitzvah.]
What is Dancing?
There is obviously a difference between walking around in a circle and dancing. The Toras Shabbos [O.H.339-2] based on a Yerushalmi defines dancing: when both feet are simultaneously in the air or not fully planted on the floor. [which would make running or jogging prohibited because its dancing on Shabbos or Yom Tov]
Clapping not Associated with Music
Only clapping connected to song would be included. Clapping, snapping or banging to get another's attention, wake someone up or silence an audience would be permitted [OCh 338 Mishna Berura 2, 4]. Clapping for applause is permitted [Oz Nidbiru Vol 13-14].
In an Abnormal Fashion
Clapping and banging was only forbidden in a normal fashion. If done in an abnormal way, for instance clapping against the back of one's hand, this would be permitted, as it would serve as a reminder, and would not lead to fixing any instruments. [O.H. 339-3]