The law of blowing Shofar when Rosh Hashanah falls on Shabbos:
One does not blow Shofar on Rosh Hashanah that falls on Shabbos.
Is the Shofar Muktzah on this Shabbos? The law regarding moving the Shofar on this Shabbos [on which Rosh Hashanah coincides with] is similar to the law of moving a Shofar on all the Shabbosim of the year, [in which the law is that] it may only be moved for the sake of [using] its space or for the need of using it [for a permitted purpose], such as to lean a plate on it and cases of the like.
May children blow Shofar on this Shabbos? A child may not blow Shofar on this Shabbos even if he has not reached the age of Chinuch. Thus, if a Beis Din hears a child blowing Shofar on Shabbos they must stop him from doing so.
Wearing a new garment on the second day: When the first day of Rosh Hashanah falls on Shabbos, in which case the Shofar was not blown on the first day, there is no need to wear a new garment during the blows of the second day.
The sages decreed against blowing the Shofar on Shabbos Rosh Hashanah. The Shofar is categorized as Muktzah Machmas Issur and thus may only be moved if one needs its space, or if one wants to make a use of it.
If one transgressed and blew the Shofar on Shabbos, has he fulfilled the Biblical Mitzvah?
In the above case, is Shehechiyanu recited by the blowing of the next day?
According to the Ashkenazim this has no relevance and the blessing of Shehechiyanu is to be repeated the next day. However, according to the Michaber and Sephardim they are not to recite Shehechiyanu the next day.
If one began blowing on Shabbos and then realized the prohibition is he to complete the sounds?
This matter is discussed in Poskim.
The Chassidic and Kabbalistic explanation:
In the Chassidic teachings, the above Rabbinical decree which prohibits the Shofar from being blown when Rosh Hashanah falls on Shabbos is questioned in terms of why the sages would make such a decree. While it is clear that the Torah gave them authority to do so, the reason explained behind the decree seems very weak, and unjustified. As stated above, the entire worry of the Sages which prompted them to make this decree was the fact that perhaps an ignoramus will come to carry the Shofar on Shabbos in an area that is a Reshus Harabim, and Biblically forbidden to be carried. Now, why would the Sages abolish a positive command ion the Torah simply due to worry of the ignoramuses. Why should all the righteous of the Jewish people lose out from a Mitzvah simply due to chance that an uneducated Jew may sin? This is even more puzzling regarding the Mitzvah of Shofar which carries such importance Divine significance. The Talmud itself explains that we crown G-d as king on Rosh Hashanah through blowing the Shofar, and in Kabbalah it is explained how the entire Divine energy for the coming year is affected by the blows. How then can such an important and significant Mitzvah be abolished for such a weak reason? The explanation is as follows:
The effect that the Shofar has above in heaven is that it arouses the pleasure that G-d has with his monarchy and with his reign over the Jewish people, thus revealing a new set of Divine energy for the coming year. Now, when Rosh Hashanah falls on Shabbos, that pleasure which needs to be aroused within G-d for Him to desire the coronation is automatically aroused by Shabbos itself. Every Shabbos Hashem experiences pleasure for Kelal Yisrael, and when Rosh Hashanah falls on Shabbos this pleasure alone suffices to renew the coronation. Thus, since the effect of the Shofar is not needed on Shabbos, therefore the Sages felt that it could be differed and abolished even due to a weak reason of that it may cause an uneducated Jew to stumble, as in any event its resultant accomplishments are already there.
 Admur 588:4; Michaber 588:5; Mishneh Rosh Hashanah 29b
 The reason: Blowing the Shofar on Shabbos or Yom Tov is not a Melacha [Admur ibid; Levush 588; Rosh Hashanah 29b;] and is only Rabbinically forbidden to be done on Shabbos and Yom Tov [Admur ibid; Rambam 2:6] due to it being a mundane act [Uvdin Dechol]. [Admur ibid; Implication of Gemara and Rashi ibid; Rif and Ran 1b] Thus, seemingly it should be permitted to be blown on Shabbos Rosh Hashanah, being it is a minor Rabbinical prohibition versus a Biblical command, just like it is allowed to be blown every Yom Tov of Rosh Hashanah due to it being a minor Rabbinical prohibition and not a complete Shvus. [If, however, the Sages would have made the blowing of a Shofar on Yom Tov a complete Shvus then even on Rosh Hashanah it would be forbidden to be blown being it is forbidden to transgress even a Rabbinical command for the sake of blowing Shofar.] Nevertheless the Sages decreed that it may not be blown on Shabbos Rosh Hashanah since everyone is obligated to hear Shofar and not everyone is expert in how to blow it there is worry that one may carry the Shofar in his hand on Shabbos and bring it to an expert to be taught how to blow and he will hence transgress the Biblical command of carrying four Amos in a public domain. [Admur ibid; Rambam ibid; Gemara ibid; M”A 588:4; Taz 588:5]
Why did the Sages not differ the day of R”H to after Shabbos in a case that it falls on Shabbos? When R”H falls on Sunday, Wednesday or Friday the Sages pushed it off to a different day, as explicitly stated in 428:1. The reason for pushing it off from Sunday is because otherwise the date of Hoshanah Raba can fall on Shabbos. Thus, so too they should have pushed it off from Shabbos in order so we can blow Shofar on both days? The reason for this is because this would cause R”H to be pushed to Monday and there is at times too many days between Monday and the Molad. Alternatively, since anyways the Shofar will be blown the next day the Sages did not feel such a need. [Beis Yosef in name of Kol Bo; Kaf Hachaim 588:33; See Levush 588:5]
 Admur 588:5; See also: Admur 308:20; Rama 308:4; Hgahos Ashri 2:24
Other Opinions: Some Poskim rule that in today’s time a Shofar is considered MMC”K. [Piskeiy Teshuvos 308:8; 588:2] Vetzaruch Iyun as what changed in this regard between then and now.
 Admur 588:6; 343:1 as explained in background
Background from 343:1: In 343:1 Admur rules that if one who sees a child under the age of Bar or Bas Mitzvah transgressing a prohibition one does not need to reprimand or stop the child from doing so, if the child is doing so for his own benefit, such as eating Treif foods or desecrating Shabbos for his own needs. Furthermore, even a Rabbinical court does not have the requirement to stop or reprimand him. However in a case that the transgression of the child will become public knowledge and there is suspicion that this may lead people to mistakenly think that what the child has done is in truth permitted being that no one protested, then the Beis Din is obligated to protest their actions even if they are doing so for their own benefit. For example, if the child was blowing Shofar on Shabbos Rosh Hashanah then since people hearing the Shofar may think that one is indeed meant to blow even on Shabbos Rosh Hashanah, therefore the child is to be reprimanded. [Admur ibid]
 Vetzaruch Iyun regarding if others including the father is obligated to protest the child’s actions. See Admur 334:26; 362:17; Ketzos Hashulchan 147:1.
 Admur 600:7; Michaber 600:3
 The reason: As the blessing of Shehechiyanu has never yet been recited over the blowing. [ibid]
The law if one by mistake blew Shofar on Shabbos with Shehechiyanu: According to the Ashkenazim this has no relevance and the blessing of Shehechiyanu is to be repeated the next day. [Kaf Hachaim 600:9] However according to the Michaber and Sefaradim he is not to recite Shehechiyanu the next day. [Peri Haretz 2:10; Birkeiy Yosef 600:4; Kaf Hachaim ibid]
 Rav Akiva Eiger brought in Piskeiy Teshuvos 588:2; Avnei Nezer Yoreh Deah 1:141; Implication of Chochmas Shlomo; Roshei Besamim 144; Ketzei Hamaeth 588
 Toras Chaim 588; See also Mahrshag 1:36; Har Tzvi 2:88
 Kaf Hachaim 600:9
 Peri Haretz 2:10; Birkeiy Yosef 600:4; Kaf Hachaim ibid
 See Chochmas Shlomo 588 and Toras Chaim 588
 See Likkutei Torah Derushim Lerosh Hashanah page 56-57