When do you roll a d20 that is not an ability check, attack roll, or saving throw?

Question

I am looking for a list of all the times in the rules you are asked to roll a d20, but the roll is not an attack roll, saving throw, or ability check.

Answer 1

The short answer: only attack rolls, ability checks, and saving throws are d20 tests.

The definition for a d20 test is given in the Character Origins glossary:

The term d20 Test encompasses the three main d20 rolls of the game: ability checks, attack rolls, and saving throws. If something in the game affects d20 Tests, it affects all three of those rolls.

The DM determines whether a d20 Test is warranted in any given circumstance. To be warranted, a d20 Test must have a target number no less than 5 and no greater than 30.

The short answer to the question here is "only those features which are attack rolls, ability checks, and saving throws are d20 tests". Ergo, features that are not those things are not d20 tests. An important defining feature of a d20 test is that it always has a target DC determining success or failure. A d20 roll which lacks a defined DC is probably a sufficient condition for not being a d20 test.

For example...

Wait, who goes first?

I’m the case of a tie for an initiative roll, the DM can call for a roll-off:

Optionally, the DM can have the tied characters and monsters each roll a d20 to determine the order, highest roll going first.

Blink

The spell blink states:

Roll a d20 at the end of each of your turns for the duration of the spell.

I cast fireball…eventually.

When under the effect of slow attempting to cast a spell with casting time 1 Action requires a roll of the d20:

If the creature attempts to cast a spell with a casting time of 1 action, roll a d20. On an 11 or higher, the spell doesn't take effect until the creature's next turn, and the creature must use its action on that turn to complete the spell.

Mirror Image

The spell mirror image states:

Each time a creature targets you with an attack during the spell's duration, roll a d20 to determine whether the attack instead targets one of your duplicates.

While this roll is determining if an attack hits, it is not itself an attack, so it is not a d20 test.

The Lucky feat?

The Lucky feat states:

Whenever you make an attack roll, an ability check, or a saving throw, you can spend one luck point to roll an additional d20. You can choose to spend one of your luck points after you roll the die, but before the outcome is determined. You choose which of the d20s is used for the attack roll, ability check, or saving throw.

Lucky tells us that when you make a d20 test, you can roll an additional d20. This additional d20 is not itself a new d20 test. Ergo, a feature giving advantage on a d20 test would not allow you to roll two additional d20s with the lucky feat. Lucky provides a method of using a non-d20-test d20 roll to modify the outcome of a d20 test.

There is also a magic item called the Potion of Possibility which mimics the effect of Lucky.

Portent

The Divination Wizard's Portent feature states:

Starting at 2nd level when you choose this school, glimpses of the future begin to press in on your awareness. When you finish a long rest, roll two d20s and record the numbers rolled.

The number rolled for Portent is used as the result for a d20 test, but the roll itself is not a d20 test. Features affecting d20 tests would not affect the initial portent roll, but they would affect the d20 test that roll result is later used for.

Wild Magic Surge

The Sorcerer's Wild Magic Surge states:

Once per turn, the DM can have you roll a d20 immediately after you cast a sorcerer spell of 1st level or higher. If you roll a 1, roll on the Wild Magic Surge table to create a magical effect.

Hit points for Gargantuan monsters

We see in the intro to the Monster Manual:

A monster’s size determines the die used to calculate its hit points, as shown in the Hit Dice by Size table.

On that table, gargantuan monsters use a d20 for calculating their total hit points. The d20s rolled for this are not d20 tests.

Taking fire damage while wearing a Helm of Brilliance.

The Helm of Brilliance states:

Roll a d20 if you are wearing the helm and take fire damage as a result of failing a saving throw against a spell.

Amputation via Sword of Sharpness

The Sword of Sharpness reads:

When you attack a creature with this weapon and roll a 20 on the attack roll, that target takes an extra 14 slashing damage. Then roll another d20. If you roll a 20, you lop off one of the target's limbs, with the effect of such loss determined by the GM.

This additional d20 rolled to see if you cut off a limb is not a d20 test.

Recharging a Cube of Force

A Cube of Force regains 1d20 charges each dawn.

Expending the last charge of a magic item

Many magic items with charges can break if you expend the last charge and feature some text similar to this line from the Wand of Magic Missiles:

If you expend the wand's last charge, roll a d20. On a 1, the wand crumbles into ashes and is destroyed.

I want my coffee, and I want it now.

Strixhaven: A Curriculum of Chaos introduced a magical coffee thermos with a d20 roll:

Each time you drink the coffee, roll a d20. On a 1, the bottle refuses to dispense coffee for the next hour.

Zombie Strahdpocalypse

Curse of Strahd features a special type of zombie called a Strahd Zombie with a limb-severing mechanic:

Loathsome Limbs. Whenever the zombie takes at least 5 bludgeoning or slashing damage at one time, roll a d20 to determine what else happens to it

Rolling on Tables: Making Magic Items Interesting

There are two tables in chapter 7 of the Dungeon Master's Guide that help the DM give some back story to magic items, and deciding these background features randomly calls for rolling a d20.

Rolling on Tables II: Caribou Boogaloo

Xanathar's Guide to Everything has several random encounter tables, and many of the possible encounters call for rolling a d20. My favorite is:

A herd of 3d20 + 60 caribou (deer) moving through the snow

Answer 2

I have tried to include every instance, this includes some that are not made by the player (eg gargantuan hit die), are not made during what we usually think of as gameplay (eg guild business background), or are made as part of one of the above rolls but are not themselves one of those rolls (eg advantage).

There is a lot of ground to cover so this will take a while to become a complete answer.

Basic Rules

• Advantage and Disadvantage Sometimes a special ability or spell tells you that you have advantage or disadvantage on an ability check, a saving throw, or an attack roll. When that happens, you roll a second d20 when you make the roll.
• Blink Roll a d20 at the end of each of your turns for the duration of the spell. On a roll of 11 or higher, you vanish from your current plane of existence and appear in the Ethereal Plane
• Mirror Image Each time a creature targets you with an attack during the spell’s duration, roll a d20 to determine whether the attack instead targets one of your duplicates.
• Slow If the creature attempts to cast a spell with a casting time of 1 action, roll a d20. On an 11 or higher, the spell doesn’t take effect until the creature’s next turn, and the creature must use its action on that turn to complete the spell. If it can’t, the spell is wasted.
• Cube of Force The cube starts with 36 charges, and it regains 1d20 expended charges daily at dawn.
• Helm of Brilliance Roll a d20 if you are wearing the helm and take fire damage as a result of failing a saving throw against a spell. On a roll of 1, the helm emits beams of light from its remaining gems. Each creature within 60 feet of the helm other than you must succeed on a DC 17 Dexterity saving throw or be struck by a beam, taking radiant damage equal to the number of gems in the helm. The helm and its gems are then destroyed.
• Many magical Staves If you expend the last charge, roll a d20. On a 1, the staff becomes a nonmagical quarterstaff.
• Many magical Wands If you expend the wand’s last charge, roll a d20. On a 1, the wand crumbles into ashes and is destroyed.
• Initiative ties If a tie occurs, the DM decides the order among tied DM-controlled creatures, and the players decide the order among their tied characters. The DM can decide the order if the tie is between a monster and a player character. Optionally, the DM can have the tied characters and monsters each roll a d20 to determine the order, highest roll going first.
• Lucky When you roll a 1 on the d20 for an attack roll, ability check, or saving throw, you can reroll the die and must use the new roll.
• Guild Business You can select your guild business from the Guild Business table or roll randomly.
• Gargantuan Hit Dice A monster's size determines the die used to calculate its hit points, as shown in the Hit Dice by Size table. (table shows "Monster Size: Gargantuan, Hit Die: d20")

Player's Handbook

• Wild Magic Surge Once per turn, the DM can have you roll a d20 immediately after you cast a sorcerer spell of 1st level or higher. If you roll a 1, roll on the Wild Magic Surge table to create a magical effect.
• Portent & Greater Portent When you finish a long rest, roll two d20s and record the numbers rolled.
• Lucky Whenever you make an attack roll, an ability check, or a saving throw, you can spend one luck point to roll an additional d20.

Dungeon Master's Guide

• Color Pools d20 table
• Psychic Wind d20 table
• Ether Cyclones d20 table
• Feywild Time Warp d20 table
• Dungeon Goals d20 table
• Wilderness Goals d20 table
• Adventure Villains d20 table
• Adventure Patrons d20 table
• Event-Based Goals d20 table
• Moral Quandaries d20 table
• Random Encounters If you roll, do so with a d20. If the result is 18 or higher, a random encounter occurs.
• NPC Appearance d20 table
• NPC Talents d20 table
• NPC Mannerisms d20 table
• Villain’s Methods d20 table
• Adventure Environments a series of 24 d20 tables
• Carousing You are caught up in a whirlwind romance. Roll a d20. On a 1–5, the romance ends badly. On a 6–10, the romance ends amicably. On an 11–20, the romance is ongoing. ... You earn modest winnings from gambling. You recuperate your lifestyle expenses for the time spent carousing and gain 1d20 × 4 gp.
• Who Created [the item] or Was Intended to Use It? d20 table
• What Minor Property Does [the item] Have? d20 table
• Chase Complications Each participant in the chase rolls a d20 at the end of its turn. Consult the appropriate table to determine whether a complication occurs.
• Side Initiative Under this variant, the players roll a d20 for their initiative as a group, or side. You also roll a d20. Neither roll receives any modifiers.
• Lingering Injuries d20 table
• Random Dungeons tables a series of 21 d20 tables

Strixhaven

• Bottle of Boundless Coffee Each time you drink the coffee, roll a d20. On a 1, the bottle refuses to dispense coffee for the next hour."