Most spells/abilities only allow the targeting of creatures rather than objects.
Of the few that do allow object targeting, most explicitly except "worn or carried" objects.
It ignites flammable objects in the area that aren’t being worn or carried. (Fireball)
A nonmagical object that isn't being worn or carried also takes the damage if it's in the spell's area. (Shatter)
What methods do exist to damage or destroy objects that are worn or carried?
I'm looking for spells, abilities, items, or monsters that could achieve this.
There are a few
We can start with the most available one: Just attack it. There are no rules against attacking objects that are worn or carried, but you would likely be subject to some DM rulings. Therefore any ability that just lets you make an attack (such as the Attack action) without a target stipulation qualifies. This also includes the attack(s) of virtually every monster. I won't bother to list all of these.
Monsters: Corrosive oozes and rust monsters
Black Pudding (MM 241) and Gray Ooze (MM 243) each have an ability which corrodes nonmagical worn/carried armour and weapons. These abilities give a -1 penalty to the item upon hit (either by the ooze or the wearer depending) and destroys them upon sufficient penalty. The abilities vary; Black Puddings can corrode weapons of wood or metal as well as any nonmagical armour, while Gray Oozes only corrode metal weapons and armour.
Rust monsters (MM 262) - while not oozes, but rather cute, insect-like monstrosities - have a similar feature which corrodes nonmagical weapon that hit it, and using their Antennae attack can corrode a "nonmagical ferrous metal object" worn or carried by a creature (that creature can make a Dex save to avoid the effect) which includes both armour, weapons, as well as any (metal) object on that person/creature.
The Disintegration ray of Beholders and Death Tyrants (and some other of their kin) can target objects (no mention of being carried or worn) and will destroy nonmagical ones (Large or smaller or 10 ft cube of Huge or larger). This is maybe one of the least scary things a beholder can do to you.
Spells: Damage to objects
As established by this answer a spell which can deal damage to objects and does not stipulate the target to not be carried, can damage them. They may be subject to the same DM rulings as attack, though. This includes the following:
Disintegrate similarly can target an object and will destroy nonmagical objects that are Large or smaller. Alternatively, you could target the creature that wears the object(s) and destroy it and all the things it carries:
This may require favourable ruling from your DM, but this 4th level spell lets you transform objects into other objects with no requirements for them not being carried or worn. However, the targeted object(s) must be considered a "raw material,” and the spell has a casting time of 10 minutes so your use cases might be limited.
Wish can do anything you can convince and phrase so your DM lets it happen. That will include interacting with carried/worn objects. I hope that interaction is worth it for a 9th level spell and enduring the stress of the free form wish. I mention wish not because it is interesting, but for completeness.
Honourable mention: Telekinesis
This spell doesn't damage or destroy objects in and of itself, but it can be used to that end. The relevant section of the spell description:
The mage hand cantrip may also be able to accomplish this, but does not have explicit rules allowing it to.
While they don’t damage or destroy the object itself, you might be able to use mage hand or telekinesis to ensure the targeted object is no longer carried or worn, and thus can be damaged by other spells like shatter or fireball.