The Torah leaders of Yerushalayim were themselves uncertain how to act. Rabbi Shmuel Salant, zt”l (1816-1909), Rav of Yerushalayim, said that if people were to ask him, he would advise them not to blow shofar. However, he would not object if people decided of their own volition to do so. It is told that the Aderes himself said that if he knew of a minyan in which shofar was blown on Shabbos, he would hide behind the wall to listen. One year, Rabbi Shlezinger did, in fact, did blow shofar – but in private so as not to generate controversy. However, on subsequent RoshHashanas that fell on Shabbos, he did not do so.
One of the main arguments against Rabbi Shlezinger is that we generally must look toward the Torah leaders of previous generations in cases of halachic uncertainty. In Yerushalayim there lived such noted Torah giants as the PriChadash, the MaharitAlgazai, and the holy OrHaChaim. None of them blew shofar when Rosh Hashanah occurred on Shabbos (Yeshurun 1, pp. 433-440).
The Geniza Discovery
In 1896, the Cairo Geniza was discovered, in which a thousand-year-old siddur was found. In it is a piyut for Maariv of Rosh Hashanah that falls on Shabbos. It describes how the shofar is blown in beis din under the strictest care so that no prohibition be transgressed in the process. From the piyut we learn that the shofar was attached to a pillar in order to ensure that it would not be carried into the public domain. The person who blew the shofar would not even touch it with his hands, but would simply place his mouth against it and blow (see Yeshurun ibid, pp. 442-443).