Objects that are not suitable for use by human beings, but are suitable for animals, do not have the status of Mukse on Shabbat, and thus may be handled. A common example of such an object is soft bones, such as chicken bones. Although people cannot make use of bones, they are suitable for dogs. Therefore, in a place where there are dogs, chicken bones do not have a status of Mukse, and may be handled on Shabbat.
However, an object that can be used by neither people nor animals does, indeed, have the status of Mukse. A common example of this kind of item is an eggshell, which has no value to humans or animals, and is therefore consider Mukse. In fact, Rav Haim Palachi (Turkey, 1788-1869) held that it is forbidden to peel a boiled egg on Shabbat, unless one does so in a manner that does not require handling the pieces of shell, meaning, the egg comes out of shell, rather than pieces of shell being removed. Hacham Ovadia Yosef disputes this ruling, noting that there is no indication in the Talmud or later authorities that peeling an egg is forbidden on Shabbat. Certainly, however, after an egg is peeled, the shells have the status of Mukse and may not be handled. It is therefore advisable to peel an egg directly over a garbage can. If one peels an egg over a counter, for example, and the shells fall onto the counter, it would be forbidden to remove them from the counter until after Shabbat.
Another concern involving eggshells is the prohibition against rendering a functional utensil Mukse on Shabbat. If one peels an egg over an empty utensil, such as a bowl or dish, the utensil becomes Mukse by virtue of its serving as a "base" for a Mukse object. Halacha forbids rendering a usable item Mukse on Shabbat, and it is therefore forbidden to peel an egg over an empty utensil on Shabbat. If one wishes to peel an egg over a utensil, he should first put some food into the utensil. This way, when the eggshells fall into the utensil, the utensil will be a "base" for both permissible and forbidden objects, and thus does not become Mukse. As mentioned, one can simply peel the egg over a trashcan, and thereby avoid all Mukse-related problems involving the shells.
Summary: Chicken bones are not considered Mukse on Shabbat, because they are suitable for dogs. Eggshells, however, are not usable for people or animals, and are therefore considered Mukse on Shabbat. One may peel an egg on Shabbat, but after peeling the egg he may not handle the eggshells. Furthermore, one may not peel an egg onto a functional utensil unless there is already something usable (such as food) in the utensil. It is advisable to peel boiled eggs directly over a trashcan, in order to avoid all Mukse-related problems.
