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.
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 Sefaradim 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.
 Blowing the Shofar on Shabbos or Yom Tov is not a Melacha and is only Rabbinically forbidden to be done on Shabbos and Yom Tov due to it being a mundane act [Uvdin Dechol]. 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. [ibid]
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 of 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]
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.
 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. [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.
 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