Does the Shofar have the Din of a כלי שמלאכתו לאיסור on the second day of Rosh Hashanah, when the first day is Shabbos, being that it is so during twilight of the second day?
The Rama in 308:4 says that a shofar is a kli shemelachto lissur and can only be moved if you need to use it
ושופר אסור לטלטלו כ"א לצורך גופו או מקומו (הגהות אשירי סוף פרק במה מדליקין):
It is forbidden to move a Shofar unless for the sake of using it or for using its place. (Ha'gahot Ashiri, end of Perek Bameh Madlikin
So to use it on Rosh hashana is not a problem since you are using the item itself. However, to carry it for no reason at all would be a problem, but the MB in 596:3 brings a heter that you might still need it to blow for someone who didn’t hear yet, so you can carry it:
ולענין טלטול לכו"ע שרי כל היום דהא ראוי לתקוע בו להוציא אחרים שלא יצאו עדיין:
RSZ in Shemiras Shabbos perek 20 footnote 53 says that normally we don’t allow carrying a kli shemelachto lissur on a chance that you might need to use it, however blowing shofar for someone who didn’t make it to shul us very common and therefore it is permitted.
It seems from this, that at night when there is no option of blowing to be motzei anyone, that it would be considered muktzah as a kli shemelachto lissur.
Note: some say that today a shofar is muktzah machmas chisoron kis, but RSZ says that even according to them, during Elul and Rosh hashana it reverts back to kli shemelachto lissur