Everyone incorrectly "trumpets" the excuse that the shofar is not sounded on Shabbat since you can't carry a shofar more than 4 amot in the public domain on Shabbat. But how many people carry their talit and prayer books all the way to synagogue?? Nearly everyone... so what's the big deal about a shofar??
Most of us should be stunned to read the the Shofar was sounded in the Temple on Shabbat:
When the Festival of Rosh Hashana falls on Shabbath, the Shofar was blown in the Beth HaMiqdash [the Temple], but not in the rest of the country. (Mishnah Rosh Hashana4:1; Talmudh Bavli 29b)
The Jerusalem Talmud, also known as the Talmud Yerushalmi asks the following question: "If blowing the Shofar is Torah-mandated, why should it not override Shabbath everywhere? And if it is not Torah-mandated, why does it override Shabbath in the Temple?"