Keep in mind that I practice in Virginia. Your state rules about Personal Property, as it relates to home sales, may vary.
I recently sold a townhouse. The day before closing, I got a note from the buyer's agent saying that the powder room mirror had been removed and they hoped it was moved for cleaning but they wanted confirmation that it would be replaced before settlement. I contacted my sellers, who had already moved out of state and they said they brought it with them because it was personal property that belonged to them and would work in their new house. When I asked why they didn't take the other bathroom mirrors, they said they had no use for them so they left them behind. As you can imagine, this set off 24 hours of stress for me the listing agent and the buyers agent, but in the end the seller was right. The mirror was hanging by a picture hook which made it personal property. The settlement went on as scheduled.
This happened one other time in my career when a seller removed a tool bench. Again, it was a seller who had moved out of state prior to closing (something to watch, I guess). This time, the buyer had to be made whole with a credit since the bench had been affixed to the wall. The distinction between personal property and a fixture comes down to whether or not it is attached to the house. In the example of the mirror, there was no difference here between removing this mirror or taking a painting. Both are hanging on the wall and not affixed. The tool bench was screwed into drywall, so it was deemed to be affixed.
To protect buyers and provide clarity to sellers, it is important to specify anything that you think should remain but which could walk during the moving process. Ideally, this would be in the initial contract, but if something is overlooked at the time of the contract, an addendum could still protect the buyer.
As a listing agent, I always discuss things like chandeliers and items that might have personal value being removed, but I will now add mirrors to my list. My rule of thumb is if you are going to take it with you, remove it and replace it prior to listing.
In the case of the mirror, the buyer told us, through their agent, that they thought the bathrooms were overdecorated for their taste (as a negotiating tactic, I assume) and that the seller could remove the chandeliers, if they so desired. I suspect my sellers thought this meant they were free to remove any of the 'fussy' elements that the buyer didn't like. There was no mal intent here...just misunderstanding.
In any transaction, clarity simplifies the process. When in doubt, write items into the contract. It is better to have these discussions up front and not at the closing table!