When one removes the shells of pistachio nuts or eggs on Shabbat, the shell becomes Mukse and may not be handled. Nutshells and eggshells are not suitable for consumption even for animals, and, as such, they are considered Mukse on Shabbat. How, then, should one discard eggshells and nutshells on Shabbat?
If one removes the shell and holds it in his hand, then he may certainly take it to the trashcan or to wherever he wishes to discard it. However, according to some authorities, including the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) and Hacham Ovadia Yosef, in his Halichot Olam, it is forbidden to place nutshells or eggshells on an empty plate. When one places a Mukse item into a utensil, the utensil becomes a "Bassis" ("base") to a Mukse item, and thus it itself becomes Mukse. The Halacha of "Mebatel Keli Me’hikano" forbids making a permissible item Mukse during Shabbat. Therefore, according to these authorities, one should not place shells on an empty plate. One should ensure that there is something else on the plate – such as a piece of bread, a piece of cutlery or a fruit – on the plate before placing the shells, as this way, he does not make the plate Mukse. The plate becomes a "Bassis" for both a permissible item and a Mukse item, and the permissible item is more valuable and significant than the Mukse item, such that the plate is deemed primarily a "Bassis" for the permissible item.
It suffices to place just a piece of bread on the plate. There is one view requiring placing a Ke’zayit of bread on the plate before the shells, but Hacham Ovadia Yosef rules that this is unnecessary. So long as the piece of bread is something significant, and not just some crumbs, it suffices to prevent the plate from becoming Mukse.
If one forgot to place something else in the plate before placing the shells, he may place something in the plate afterward. The plate then becomes permissible for handling, and thus it may be carried and discarded into the trash.
Interestingly, Rabbi Moshe Halevi (Israel, 1961-2001), in his work of responsa Tefila Le’Moshe, rules more leniently in this regard. He contends that since the shells are valueless, they are subordinate ("Battel") to the plate and cannot render it Mukse. In his view, then, it would be entirely permissible to place nutshells or eggshells into an empty plate on Shabbat. One who wishes to rely on this opinion may do so, though it is preferable to follow the stringent position and ensure to first place a permissible item into the plate before placing the shells.
Finally, it should be noted that one should ensure before discarding shells into the trashcan that the can is not empty. If the can is empty, then one will render it Mukse by throwing the shells into it. For this reason, there are people who make a point of ensuring when Shabbat begins that there is something significant in the trash can, so that they will not violate the prohibition of "Mebatel Keli Me’hikano."
Summary: After one peels an egg or removes the shell of a nut, the shell becomes Mukse. If the shell is still in his hand, he may discard it wherever he likes. According to some views, one should not place shells in an empty plate, trashcan or other utensil, and it is proper to follow this stringent ruling. One should therefore ensure that there is something else in the plate or trashcan before placing the shells in it.
