Here is something that I suppose I should have known, but "discovered" today. If you are trying to remove a Property Set from multiple objects and your current selection includes at least one object that does not have that Property Set attached, you will not be offered that Property Set as a choice for removal when you choose the Remove property sets button on the Extended Data tab of the Properties Palette.
This does not apply to the reverse of this situation. If you are trying to add a Property Set to multiple objects (to which the Property Set applies) and happen to include an object that already has that Property Set attached, you will still be offered the option to attach the set.
This came to my attention because there was a drawing file (done by others) that had duplicates of two of our standard Property Set Definitions (one for Spaces and one for Doors) and the duplicates were not purgeable. The duplicates were not referenced in a Schedule Table Style, and when selecting all of the Doors or all of the Spaces in the drawing, the duplicates were not on the list that could be removed when using the Remove property sets button on the Extended Data tab of the Properties Palette, giving the impression that they were not attached.
I created two Schedule Table Styles (one for Spaces and one for Doors) to identify whether any objects had these sets attached, each using at least one property from the associated duplicate Property Set Definition. I placed an instance of each Schedule Table in the drawing, selecting all objects for the selection set for each table. Objects with ?? in the columns from the duplicate Property Set did not have it attached; objects with real data did. It turned out that there was only one Space with the duplicate Space set, which would have been tedious to find manually. A large number of Doors had the duplicate Door set, so I used the Add All Property Sets choice from the Schedule Table context menu to add the duplicate Property Set to all Doors, and was then able to use the Properties palette to remove that set from all Doors. After deleting the Schedule Tables and purging their definitions, I was able to purge the duplicate sets from the drawing.
As to how those duplicate sets got in the file and attached to objects in the first place, that will have to remain a mystery.