Covering the Shofar
There is a widespread custom for the baal tokea, the one who blows the shofar on Rosh Hashana, to cover the shofar with a tallit, or other covering, when reciting the preliminary blessings before it is blown. If this is done, the baal tokea should hold the shofar under the tallit while he recites the blessings. There are also those who briefly cover the shofar between each set of shofar blasts.
On the other hand, there are those who argue that the shofar should be exposed and openly held in one's hand when the blessings are recited. According to this approach, the shofar is considered to be similar to food and other ritual items that must be visible when reciting a blessing upon them. Nevertheless, the more widespread custom is to cover the shofar when reciting the blessings.
It is actually unclear where the custom to cover the shofar originates from. Some suggest that covering the shofar is intended to recall the Midrashic teaching that Avraham Avinu covered Yitzchak while he prepared the altar for the Akeida. He did this so that the Satan would not be able to drop a rock on Yitzchak, or otherwise injure him in any way, which would disqualify him as an “offering.” This is because anything blemished may not be offered as a sacrifice. Alternatively, covering the shofar, the ram's horn, serves to recall the actual ram that was offered in place of Yitzchak which was "hidden" from Avraham until it was needed at the end of the Akeida episode. It is also taught that the shofar is covered so that the Satan will not realize that we are about to blow the shofar, and therefore try to prevent us from doing so.
It is also suggested that covering the shofar is intended to save it from “embarrassment,” similar to the custom of covering the challah before Kiddush. This is because the blessing recited upon the shofar does not actually refer to the shofar itself, but rather "…lishmoa kol shofar," merely to its sound. Finally, some say that the custom to cover the shofar originated at a time when the authorities banned Jews from blowing the shofar. Hence, it was covered so that it would remain out of sight for as long as possible.
In addition to the custom of covering the shofar when the blessings are recited, there is also a popular custom for the baal tokea to designate and prepare a number of additional “emergency” shofars. When this is done, all the shofars are placed under the tallit (or other covering) before the blessings are recited. The reason for this practice is that should the baal tokea be forced to use a different shofar(s) during the course of the service, the replacement shofar(s) will have been “included” in the original blessings and thereby “activated” for use if needed, as well. Similarly, in the event that the baal tokea is unable to blow the shofar or must leave before the blowing is completed, any other person who was present when the shofar blessings were recited may take his place. This is because the blessings recited by the original baal tokea allow anyone else to blow the shofar on behalf of the congregation, as well.
 Kitzur Shlah; Be'er Heitev, OC 593:3; Mateh Ephraim 585:3; Elya Rabba 585:8; Siddur Yaavetz 35.
 Siddur Yaavetz; Ben Ish Chai, Nitzavim 15; Kaf Hachaim, OC 585:14.
 Kaf Hachaim, OC 585:14.
 Rivevot Ephraim 4:144:6.
 Elef Hamagen 585:6; Minchat Elazar 4:36.
 Tzitz Eliezer 7:27:6; Minhag Yisrael Torah, OC 585:6.
 Bereishit Rabba 56:5; Elef Hamagen 585:7; Be'er Heitev, OC 593:3.
 Vayitzbor Yosef 18.
 Magen Avraham 585:11; Likutei Maharich; Chelkat Yaakov, OC 222 (3:128 in some editions).
 Avnei Nezer 431.
 Minchat Elazar 36, Divrei Yatziv, Hashmatot 56; Vayitzbor Yosef 18.
 Elef Hamagen 585:6.
 OC 585:3.