The Poskim discuss the status of various items in terms of Hilchot Mukse. Each item must be analyzed and classified in the proper category to determine what its restrictions are.
A Shofar is classified as a Keli She'm'lachto L'isur, a vessel whose primary purpose is prohibited on Shabbat. Accordingly, if it was in the way and the space it occupied was needed, it may be removed. According to Hacham Ovadia, the Shofar retains its Mukse status even on Rosh Hashana that fell on Shabbat. Since Shofar is not blown in such an event, it is still a Keli She'm'lachto L'isur. On Rosh Hashana that falls on a weekday, there is clearly no issue of Mukse with the Shofar; it may be handled as needed, to be blown as many times as necessary.
Each of the Four Species must be analyzed separately to classify its Mukse status. The Lulav branch itself may obviously be handled to perform the Misva or to give it to someone, including a woman, to perform the Misva. Moreover, the Kaf Hahaim (Rav Haim Palachi, Turkey, 1788-1869) establishes a rule by which the Lulav may be handled, even after being used for the Misva, to be put back in a secure place. Therefore, it may be taken home from the Bet Knesset, out of concern that it will not be safe if left in a public place. Similarly, if he is concerned that the place he stowed the Lulav at home is potentially compromised, he may move it to a better place. However, once the Misva has been completed by all parties and the Lulav is in a secure place, it becomes Mukse. On Shabbat Succot, there is no Misva to shake the Lulav and it becomes Mukse, since it has no other use and is not a vessel.
Hadasim are generally not Mukse on Shabbat throughout the year, since their primary function is for smelling. However, on Succot, the Hadasim of the Misva have been designated exclusively for use with the Lulav and may not be smelled. Therefore, they are Mukse on Shabbat of Succot.
An Etrog is classified as food and is not Mukse on Shabbat throughout the year. On Succot, the Etrog is exclusively designated for the Misva and may not be eaten. However, the secondary use of the Etrog is for smelling, and that is not prohibited on Succot. Therefore, the Etrog is not Mukse on Shabbat of Succot.
The Shemirat Shabbat K'hilhata rules that Sisit strings that were not yet tied to the Tallit, are Mukse Mahamat Hesron Kis. That is, they have no function on Shabbat, because they are too valuable to use for any other permitted function.
Sisit strings that were torn out of a Tallit are sometimes used as a bookmark. Therefore, they do not become Mukse, because there is a use for them.
The Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933) and Rav Moshe Feinstein (Russia-New York, 1895-1986) ruled that a blank sheet of paper is Mukse Mahamat Hesron Kis. In previous generations, paper was a very precious item, and no one would use it for any purpose besides writing. However, contemporary Poskim, including Hacham Ovadia, rule that paper is a cheap item and is used for other purposes such as cleaning a spill or covering something. Therefore, it is certainly not the strictest form of Mukse. The Poskim debate whether it has a status of a Keli She'm'lachto L'heter and has no restrictions, or a Keli She'm'lachto L'isur, and would only be permitted to handle for a specific permitted function such as covering a pot or moving it out of the way. Nevertheless, special stationary paper, which is not used for other functions, retains its strict Mukse status.
Generally, a person puts his Pesach dishes away for the year, not intending to use them again until next year. If a person wanted to use some of his Pesach dishes on Shabbat during the year, are they considered Mukse? Rav Shlomo Zalman Auerbach (Israel, 1910-1995), as cited in Shemirat Shabbat K'hilhata, argues that they are permitted. He reasons that in order to become Mukse Mahamat Hesron Kis, the object must be excluded from use for any other purpose besides it's primary, prohibited purpose. For example, a Mohel's knife is Mukse Mahamat Hesron Kis, because it would never be used for anything besides performing the Brit. However, the primary function of dishes is to eat from, and that is precisely what he wants to do in this case. Their primary function is permitted. The fact that he set them aside for Pesach does not preclude them from use as Mukse. Similarly, the Halacha would not prohibit wearing a suit on Shabbat that was bought and designated to be a Yom Tob suit.
-A Shofar is Mukse on Shabbat, including Shabbat of Rosh Hashana.
-The Lulav and Hadasim are Mukse on Shabbat of Succot, but only the Lulav is Mukse on all other Shabbats. The Etrog is not Mukse on any Shabbat.
-Sisit Strings which have never been tied to a Tallit are Mukse. Sisit Strings which were torn out of a Tallit are not Mukse.
-Regular paper is not Mukse, but expensive stationary is Mukse.
-Pesach dishes are not Mukse and may be taken out of storage and used on Shabbat throughout the year.