Great question, and one that arises often in divorce.
If the decree of divorce provides for what happens to unclaimed/unpossessed property, you would rely on the court’s orders as articulated in the decree. Rarely does this happen, but read your decree to see if it did in your case.
As best I can conclude after doing much more research on this question than I thought would be necessary to answer it (and I am still not completely sure what the answer to your question is), property does not automatically acquire “abandoned” status after being left in someone else’s possession for a particular period of time.
It’s tempting to claim that the matter can be resolved by appealing to landlord-tenant law, which is quite clear on how a landlord must behave before it can dispose of a tenant’s abandoned property. The problem is that the relationship between freshly divorced people is not one of landlord and tenant.
What about claiming that the matter can be resolved by self-storage facility law (Utah has a whole set of code sections on the subject at Utah Code Title 38, Chapter 8)? No. The relationship between freshly divorced people is not one of storage customer and storage provider under self-service storage facilities law.
Researching this question has inspired me to include in a divorce settlement agreement and decree of divorce provisions for how property abandoned by one ex-spouse at the other ex-spouse’s home or other location owned by or under the control of that ex-spouse. Because divorce-specific law on this subject simply doesn’t exist (which is odd because the problem of an ex-spouse failing to take his/her property out of the house/apartment when ownership or occupancy of the house/apartment was awarded to the other spouse arises so often in divorce).
If the decree of divorce is silent on the matter of what one in possession of property left behind by his/her ex-spouse can do with that property, then you need to look to what the law governing bailments, lost and abandoned property, and conversion. There may very well be other law(s) you need to consider, but you need to be aware of at least bailment, lost and abandoned property, and conversion law.
From Black’s Law Dictionary (11th ed. 2019):
abandoned property. (1841) Property that the owner voluntarily surrenders, relinquishes, or disclaims. Cf. lost property; mislaid property.
bailment n. (16c) 1. A delivery of personal property by one person (the bailor) to another (the bailee) who holds the property for a certain purpose, usu. under an express or implied-in-fact contract. • Unlike a sale or gift of personal property, a bailment involves a change in possession but not in title. Cf. pawn.
– constructive bailment. (1843) A bailment that arises when the law imposes an obligation on a possessor of personal property to return the property to its rightful owner, as with an involuntary bailment. • For example, a police department becomes a constructive bailee for an impounded vehicle. Cf. involuntary bailment.
– involuntary bailment. (1840) A bailment that arises when a person accidentally, but without any negligence, leaves personal property in another’s possession. • An involuntary bailee who refuses to return the property to the owner may be liable for conversion. — Also termed involuntary deposit. See abandoned property, lost property, mislaid property under property. Cf. constructive bailment.
CONVERSION. 2. Tort & criminal law. The wrongful possession or disposition of another’s property as if it were one’s own; an act or series of acts of willful interference, without lawful justification, with an item of property in a manner inconsistent with another’s right, whereby that other person is deprived of the use and possession of the property. — Also termed tortious conversion; criminal conversion. — convert, vb. — conversionary, adj.
– conversion by taking. (1905) Conversion by taking a chattel out of the possession of another with the intention of exercising permanent or temporary dominion over it, despite the owner’s entitlement to use it at all times.
– conversion by wrongful delivery. (1917) Conversion by depriving an owner of goods of possession by delivering them to someone else.
– conversion by wrongful destruction. Conversion by willfully consuming or otherwise destroying a chattel belonging to another person.
– conversion by wrongful disposition. Conversion by depriving an owner of goods by giving some other person a lawful title to them.
– direct conversion. (1841) The act of appropriating the property of another to one’s own benefit, or to the benefit of a third person. • A direct conversion is per se unlawful, and the traditional requirements of demand and refusal of the property do not apply.
– fraudulent conversion. (17c) Conversion that is committed by the use of fraud, either in obtaining the property or in withholding it.
– innocent conversion. See technical conversion.
– involuntary conversion. (1876) The loss or destruction of property through theft, casualty, or condemnation.
– negligent conversion. See technical conversion.
– technical conversion. (1871) The taking of another’s personal property by one who acts in good faith and mistakenly believes that he or she is lawfully entitled to the property. — Also termed innocent conversion; negligent conversion.
In the Utah Supreme Court case of Gurgel v. Nichol (429 P.2d 47 (Utah 1976), the question was whether the seller of a house who left some personal property behind in the house had thus abandoned the property and lost ownership of the personal property as a result. Here is what the Utah Supreme Court ruled:
 Whether or not personal property is abandoned by the owner is a matter of intention, which is ordinarily a question of fact to be ascertained by a jury upon a consideration of all of the facts and circumstances in the case.
 Section 41 of 1 Am.Jur.2d, Abandoned, Lost, Etc., Property, reads as follows:
Mere nonuse of property, lapse of time without claiming or using property, or the temporary absence of the owner, unaccompanied by any other evidence showing intention, have generally been held not enough to constitute an abandonment. However, such facts are competent evidence of an intent to abandon and as such are entitled to great weight when considered with other circumstances, as, for instance, a failure on the part of the owner by acts or otherwise to assert any claim to the property or right alleged to have been abandoned. But mere nonuser is not evidence of abandonment unless it continues for the statutory period of limitation of actions to recover the right or property, or unless it would be inequitable to allow the right to be asserted.
The probative force of a showing of absence, lapse of time, and nonuser may of course be rebutted by proof of facts or circumstances explaining the relinquishment and showing the absence of an intention to abandon the thing or the right, and slight circumstances have been allowed to rebut the inference of abandonment arising from long disuse. * * *
 It may be that upon a trial of this case the jury, or the court if no jury is had, could find from a lapse of time that there had been an abandonment, but we do not believe that the court can say as a matter of law that personal property is abandoned when nothing more is shown than a delay of nine months is coming to get it.
 The plaintiff made some claim of right to the personal property by reason of the terms ‘tenements, hereditaments, and appurtenances’ in the deed given to his predecessor. This claim is dismissed as being without merit.
To treat property as abandoned it appears that (in Utah) you must establish this fact by filing a law suit (and in that law suit establish that did everything required of you, including giving the owner of the property sufficient notice) to have it declared abandoned before you can assume ownership and control of it.
If you want to play it safe (in Utah) and wait until the statute of limitations for conversion has run, it appears to me that the limitations period is 3 years. But even then, there are exceptions that can “toll” the statute of limitations, so be warned.
 Perhaps something like this (consult an attorney knowledgeable on the subject in your jurisdiction to know how to address and resolve this issue in compliance with the law):
“Husband/wife has X days from the date this Decree of Divorce is entered to retrieve his/her ________, _______, and _______ from the marital residence, and if it is not retrieved on or before X days from the date this Decree of Divorce is entered, husband/wife forfeits his/her rights to the property and [the other ex-spouse to whom the property was not awarded] shall be free to keep said property as his/her own property and/or to dispose of said property has he/she pleases.”
If you are not yet divorced and you don’t have a clause like this in your divorce settlement or decree, it may be wise to include such a clause to define when property is deemed abandoned and establishing your right to own and/or dispose of it at the time the decree of divorce is entered.
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