A utensil that broke is usually considered Muktzeh as it serves no purpose.
However, in certain circumstances, if something breaks in a way that the broken parts can be put back together at a later point, it is not Muktzeh. It would have to be something in which the primary part of the item can be used.
For example, a button that falls off on Shabbos may be moved as it will be sewn back on after Shabbos. Similarly, if the arm rest of a chair falls off, one may move the arm rest on Shabbos.