What is the Halacha about clapping on Shabbat. Does it matter if it is in shul or not?

It is forbidden to clap hands on Shabbat if not by a Shinui - on the back of ones hand. The Shulchan Aruch Orach Chayim 339 and the R"ma both tended to be maykel, in order that the people who clapped would not be commiting a sin. The reason for the Isur is so as to stop one from fixing a musical instrument on shabbat. Nowadays, when we do not know how to fix instruments there is more reason to be Matir. Rabbi Ro'i Margalit