The d3 has been used for many different mechanics.
This search on DnDBeyond gives all the instances of "d3" appearing in the DDB compendium material, which covers the first three sections here. Note, this one has several false positives, as "D3" is used to denote locations in many adventures. All the other searches I link here use "1d3" to get hits.
The d3 first shows up in the introduction to the Player's Handbook, where it is explained how you can simulate one with a d6:
The same d notation appears in the expressions “1d3” and “1d2.” To simulate the roll of 1d3, roll a d6 and divide the number rolled by 2 (round up).
We now examine many of the ways the d3 has been used.
Falling down a chimney.
In one of the adventures published in Explorer's Guide to Wildemount, you can fall down a chimney:
Any character who tries to descend the chimney without a rope or a climbing speed inevitably falls. Roll a d3 to determine which lower chimney the character tumbles down.
Betting on dinosaur racing and meeting with princes.
In Tomb of Annihilation, you can bet on dinosaur races at one of the major towns. Two of the possible results depend on the outcome of a d3 roll.
In this same town, the party can seek an audience with a merchant prince. The module states:
Characters must wait 1d3 hours before being given an audience.
Having close family relationships.
In the character building options in Explorer's Guide to Wildemount, you roll a d3 to determine how many of your family members you are close to:
Roll a d3. This is the number of powerful relationships you have within your family.
And a plethora of magic items.
This search shows 33 magic items that use a d3 in some manner, most of them are for recharging at dawn. I will not list them all here.
The Ioun Stones use a d3 to determine how far from your head the stone orbits.
The decks of Many Things and Several Things use a d3 to determine the number of times a certain ability can be used.
And a few monsters.
This search gets us the mosnters. Five monsters use a d3 to determine the number of creatures they summon with a particular ability. For example, the Crokek'toeck:
Crokek’toeck opens its mouth and disgorges 1d4 barlguras, 3d6 gnolls led by 1 gnoll fang of Yeenoghu, 6d6 dretches, or 1d3 vrocks.
The Assassin Bug lays eggs inside you:
The target is infested with 1d3 assassin bug eggs, which immediately hatch into assassin bug maggots.
The Olhydra makes it rain:
Violent downpours become frequent within 10 miles of the lair. A downpour occurs once every 2d12 hours, and lasts 1d3 hours.
How many slaves does a Booyahg Goblin have?
This search gets us the goblin playable race. But this is misleading. DDB took the Monster Lore section on Goblins from Volo's Guide to Monsters and put it on the same page with the goblin playable race information. I recommend against giving your players slaves.
Booyahg Whip. Khurgorbaeyag saw fit to gift this goblin with powers that enable it to dominate others. The goblin has 1d3 other goblins that slavishly obey its orders.
And an indeterminate amount of times for random encounters.
This search gives an overflow of results at DnDBeyond. I'm not going to count them. Most are random encounters from Xanathar's Guide to Everything, but there was a hit for Tales from the Yawning Portal, as well as one for Icewind Dale: Rime of the Frostmaiden. .