Not all weapons can be enhanced. The ones that are eligible for enhancement have a wrench icon on the up-right corner in their weapon icon. Both permanent and duration weapons can be enhanced but note that when the duration weapon expires, so does the enhancement.
This is a list of weapons that can be enhanced. Weapons listed here that have Gold and Christmas editions are enhanceable as well, though there will be no cosmetic changes.
Note that any enhanceable weapon which is stored in Family storage will not be applied.