Only creatures that use a class's spell list and have spell slots or who have the Ritual Caster feat can use Spell Scrolls.
In the published official material as of 2018, this includes:
- PCs with levels in the Bard, Cleric, Druid, Paladin, Ranger, Sorcerer, Warlock or Wizard classes
- PCs who are Arcane Trickster Rogues or Eldritch Knight Fighters
- Monsters that use a class's spell list and have spell slots (i.e. a Guardian Naga)
- Creatures with the Ritual Caster feat
And excludes PCs and monsters who do not meet the previous criterion.
You could also rule that having a feature that refers to a class's spell list in its text is sufficient but this leads to very odd scenarios where being granted the choice of picking a spell gives you access to cast spells from spell scrolls, but being given spells directly without the ability to pick one from a spell list does not. If you want to use that interpretation then the following features also let you use spell scrolls:
- High Elf subrace
- Magic Initiate feat
- Spell Sniper feat
This ruling reads somewhat differently from the written rules, but it's the only way to reconcile the rules as written with the rules as intended. Let's see how we got there:
Initial RAW reading
As you rightly point out, the requirement to cast spells from scrolls as referenced by the Spell Scroll magic item (DMG pg. 200) is that:
...if the spell is on your class's spell list, you can use [it]. Otherwise, the scroll is unintelligible...
At first this appears straightforward. It means that casting a spell from a Spell Scroll is restricted to creatures that both:
- Have a class with a spell list
- Have that specific spell on their class's spell list.
This means that the following classes qualify: Bard, Cleric, Druid, Paladin, Ranger, Sorcerer, Warlock, Wizard.
Conversely, the following classes do not: Barbarian, Fighter, Monk, Rogue
For PCs this simply means having a level in one of the qualifying classes.
What about monsters?
For monsters we can rely on the rules on attunement (DMG pg. 136-137) which states:
...a monster qualifies [as a member of a class] if that monster has spell slots and uses that class's spell list.
This means that only certain spellcasting monsters qualify. Let's first look at the Guardian Naga's Spellcasting trait:
Spellcasting: The naga is an 11th-level spellcaster. Its spellcasting ability is Wisdom (spell save DC 16, +8 to hit with spell attacks), and it needs only verbal Components to cast its Spells. It has the following Cleric Spells prepared:
- Cantrips (at will): Mending, Sacred Flame, Thaumaturgy
- 1st level (4 slots): Command, Cure Wounds, Shield of Faith
- 2nd level (3 slots): Calm Emotions, Hold Person
- 3rd level (3 slots): Bestow Curse, Clairvoyance
- 4th level (3 slots): Banishment, Freedom of Movement
- 5th level (2 slots): Flame Strike, geas
- 6th level (1 slot): True Seeing
Sure enough a Guardian Naga uses the Cleric spell list and has actual spell slots, so it qualifies as a member of the Cleric class and thus gets to cast spells from the Cleric spell list from Spell Scrolls.
Next lets look at the Innate Spellcasting trait of a Rakshasa:
Innate Spellcasting: The rakshasa's innate spellcasting ability is Charisma (spell save DC 18, +10 to hit with spell attacks). The rakshasa can innately cast the following Spells, requiring no material components:
At will: Detect Thoughts, Disguise Self, Mage Hand, Minor Illusion
3/day each: Charm Person, Detect Magic, Invisibility, Major Image, Suggestion
1/day each: Dominate Person, fly, Plane Shift, True Seeing
They make no mention of using a class's spell list and do not have spell slots. Thus they do not qualify as a member of any class and do not get to cast any spells from Spell Scrolls.
What about Arcane Trickster Rogues and the Eldritch Knight?
As stated by Jeremy Crawford and related in the question can an arcane trickster use a spell scroll from the wizard spell list?, Arcane Trickster Rogues and Eldritch Knight Fighters are intended to be able to use Spell Scrolls of the Wizard spell list, which means that they are intended to qualify as members of the Wizard class for the purpose of being eligible to use Spell Scrolls.
How can we justify this? The most straightforward method is to use the same rules we used to determine which monsters qualify as members of a class. Both Arcane Tricksters and Eldritch Knights use the Wizard spell list and have spell slots. This means both sub-classes qualify as a members of the Wizard class and thus gets to cast spells from the Wizard spell list from Spell Scrolls.
What about feats?
No feats grant you any spell slots, but as illustro points out Ritual Caster does say:
If you come across a spell in written form, such as a magical spell scroll or a wizard’s spellbook, you might be able to add it to your ritual book.
The requirement for casting spells from Spell Scrolls is tied to being able to understand what's written on the scroll. If this feat allows you to read from spell scrolls of your chosen class, then that means you must also be able to cast spells from those Spell Scrolls.
This now leaves us with 3 possible interpretation as to why Ritual Caster lets you qualify:
- A) Specific beats general: Ritual Caster says you can use spell scrolls while other feats and features that do not let you qualify do not, and that is sufficient justification.
- B) The actual requirement is tied to your ability to learn new spells using your spellcasting feature: This works for spellcasting classes, Arcane Trickster, Eldritch Knight and Ritual Caster, but not for anyone else. This works but is even more awkward to use as a guideline than the monster requirement to have spell slots.
- C) The only requirement is to have a feature that uses a class's spell list: This is the simplest option of them all but I do not like it because it means that just because the High Elf's cantrip feature lets them pick a cantrip they like from the Wizards spell list, then they get access to casting spells from that entire spell list in spell scroll form. This feels wrong because the Dark Elf's Drow Magic gives them a cantrip directly as well as 2 other spells, but since they didn't let you pick the spells from a class's spell list they can't use spell scrolls.
Option B doesn't feel very convincing and option C just feels wrong to me so I will go with option A, but you are of course free to use your favorite interpretation.
And that's about it, we've now built the full list of creatures who can cast spells from Spell Scrolls.
Traits that do not help you qualify
For the sake of completion, here are some examples of traits that do not let a creature qualify to use Spell Scrolls:
- Path of the Totem Warrior Barbarian: No spell list and no spell slots.
- Way of the Four Elements Monk: No spell list and no spell slots.
- Magic Initiate: A spell list, but no spell slots. (Qualifies with option C)
- Spell Sniper: A spell list, but no spell slots. (Qualifies with option C)
- High Elf's Cantrip: A spell list, but no spell slots. (Qualifies with option C)
- Dark Elf's Drow Magic: No spell list and no spell slots.
- Tiefling's Infernal Legacy: No spell list and no spell slots.