Review: If the part that broke is not fit for use and one plans on throwing it out, or even if it is fit for use, but most people throw it out, it is Muktzeh.
If the part that broke is fit for use and people are accustomed to keep it, it is not Muktzeh.
Generally, if the part that broke off is not fit for use but it is common to keep so that it can be reattached after Shabbos, it is not Muktzeh.
The above is only if the main part of the utensil can be easily used. If it can be used but it is uncomfortable to use (like broken glasses), one may not us it out of concern that one will fix it.
If the item was attached to the ground (or home) and broke, it cannot be moved on Shabbos.