Let's say someone makes egg salad on Shabbos and when peeling the eggs puts the peels in a cup. Is this cup allowed to be moved (even without adding anything that's permitted to be moved in the cup)?
If the shells are fit for consumption by a kind of animal that is common in the area you're in [or, I assume, that you own a specimen of], then they are not muktzim. (Sh'mira Shabas K'hilchasah 20:27.)
If, however, they're not, then they are muktzim. One should release the shells from his hand as soon as feasible after peeling the eggs (SSK 20:26), though one need not do so immediately (SSK 22:33).
SSK 22:fn38 struggles to explain why the plate or cup the shells are placed in does not become muktze, but nonetheless accepts that as fact. Thus, one can carry the plate to the trash receptacle and tip the shells out; likewise, if the shells are on the table, one can wipe them off — but without touching them. This assumes you're doing so because you need the table/plate clean or because the shells' sitting there disgusts you; otherwise, it may be forbidden to move the shells by any means. (SSK 22:36.)
(Citations to SSK are to the 5739 edition.)