Many decorate the synagogue with pretty, fragrant branches and plants, in honor of the Torah that was given on Shavu’ot. Since Torah adds life, it is customary to decorate the synagogue with plants or branches, which symbolize life. This custom also reminds us that at the time of the giving of the Torah, Mount Sinai was covered by plants to honor the Torah (Levush). Some use fragrant branches in order to increase joy and pleasure, and to represent the exquisite spirituality of the Torah: “With every statement that issued forth from God’s mouth, the entire world filled with fragrance” (Shabbat 88b). Some use tree branches, because God passes judgment on fruit trees on Shavu’ot. By seeing branches in the synagogue, people will remember to pray for the trees (MA 494:5). However, branches from fruit trees should not be used, as they should not be broken off without a good reason.
This custom began in Germany around 600 years ago (Maharil; Rema 494:3). From there it spread to most Jewish communities, both Ashkenazic and Sephardic. The Vilna Gaon opposed the custom, because he felt it was similar to gentile practices. Christians decorated their homes with branches before their holiday (Pentecost), and the Torah commands us not to imitate their practices. Thus we read: “nor shall you follow their laws” (Vayikra 18:3). Some rule in accordance with the Vilna Gaon (Ḥayei Adam 131:13).
Nevertheless, most poskim maintain that there is no problem with the custom. The prohibition of following the gentiles applies only when either the custom is immodest in some way, or it is meaningless and pointless (and is done only to imitate non-Jewish practice, which can be assumed to be based on false beliefs). In contrast, in this case there are good reasons for the custom, and there is no concern that it looks like following non-Jewish ways. Accordingly, most Jewish communities do decorate the synagogue with pretty, fragrant branches and plants. Some decorate their homes as well with branches and flowers (Rema 494:3).
Since the branches are for decoration, they are not muktzeh. However, if Shavu’ot is on a Sunday, the branches should not be set out on Shabbat. Doing so would be preparing on Shabbat for Yom Tov, which is prohibited (MB 494:9).