The Shulhan Aruch (308:4) states that a Keli She'm'lachto L'heter-A utensil which has a permitted function, may be moved for any purpose, including its own protection. However, Maran adds a caveat that even such a utensil should not be moved "Shelo L'sorech Klal"-for no purpose whatsoever. That is, one should not fidget with these items for no reason. Maran then lists two exception to this rule: food and holy books. These items may be moved randomly, in any fashion. Some Poskim want to include clothing and jewelry in this category of exceptions, as well. However, Hacham Ovadia in Hazon Ovadia (Vol. 3, p. 34) cites the Me'iri who rules that clothes and jewelry may not be moved randomly.
Hacham Ovadia (Hazon Ovadia Vol 3, p. 47) rules that a food vessel requiring Tevilah (immersion) in a Mikveh before use, is not Mukse, even though it may not be immersed on Shabbat. He bases his leniency on the fact that some opinions do permit Tevilat Kelim on Shabbat, and even according to the mainstream opinion, the vessel can be given to a non-Jew, which removes the obligation for immersion, and then be used by borrowing it back from the non-Jew.
A pot which became non-kosher, which needs Koshering through Hag'ala, is not Mukse. Hacham Ovadia explains that although it may not be koshered on Shabbat, it is still suitable for containing cold food.
There is a question whether one may move a food item which has a questionable Kashrut. For example, there is debate whether powdered milk was included in the prohibition of consuming milk from a non-Jew. Hacham Ovadia was strict on this matter and ruled that such products are not kosher. If one had a chocolate bar made with such milk, would it be Mukse on Shabbat? Hacham Ovadia gives several reasons why it is not Mukse. First, it may be given to a non-Jew or even to young children. Second, the dissenting opinions who do permit the powdered milk of non-Jews may be relied upon in the context of Mukse.
Hacham Ovadia (Hazon Ovadia Vol. 3, p. 9) rules that stale bread is not Mukse, because it can be consumed in soup on Shabbat.
Non-Mukse items may not be moved randomly, for no reason, except for food and holy books.
Pots in need of immersion or Koshering are not Mukse. Products made with non-Kosher powdered milk are not Mukse.
Stale bread is not Mukse.