When are Ability Scores (not modifiers) used in D&D 5e?

Question

In D&D, the main ability is a score from 1-30, while the modifiers range from -5 to +10 (score -10/2 round down)

In the Player's Handbook, it states:

Because ability modifiers affect almost every Attack roll, ability check, and saving throw, ability modifiers come up in play more often than their associated scores.

So when do the scores come up?

Answer 1

Pretty rarely, but there are a few examples

Strength

• Carrying capacity - The score, not the modifier is used to determine how much a character can carry without becoming encumbered.
• Armor proficiency - The score, not the modifier is used to determine if a character can properly wear heavy armors without taking penalties to speed.
• Jumping distance - The modifier controls the height of a vertical jump, but the score controls the length of a long jump.
• Used for determining whether a character can multiclass.
• Strength 13 is a pre-req for the Grappler feat.

Dexterity

• Used for determining whether a character can multiclass.
• I couldn't identify any other RAW examples, but I have played at a lot of tables which will use the characters Dex score to break ties on initiative.
• Dex 13 is a pre-req for the Defensive Duelist and Skulker feats.

Constitution

• After some digging, I found that the Gas Spore's death attack causes death based on the target's Constitution score.
• The number of minutes you can be in frigid water is tied to your Constitution score.

Intelligence

• Used for determining whether a character can multiclass.
• Movement on the Astral Plane is actually based upon a character's Intelligence score (speed=3 times their Intelligence score).
• A lot of abilities and attacks are only effective against creatures whose intelligence is greater than a certain score. Like Psychic Scream, the Neothelid's Creature Sense, the Telepathy spell, the Beast Bond spell, and Awaken.
• The School of Necromancy Wizard's feature Command Undead is heavily controlled by the target's intelligence score.
• Controls the DC of ciphers created with the Linguist feat.
• Intelligence 13 is a pre-req for the Ritual Caster feat.

Wisdom

• Used for determining whether a character can multiclass.
• Wisdom 13 is an alternative pre-req for the Ritual Caster feat.
• Controls specific elements of the Knowledge Cleric's Visions of the Past feature.

Charisma

• Used for determining whether a character can multiclass.
• Cha 13 is a pre-req for the Inspiring Leader feat.

Answer 2

Modifications work on scores rather than modifiers.

When character gains an ability score improvement, usually at levels 4, 8, 12, 16, and 19 (though the fighter and rogue get more) the characters' ability scores are increased, rather than their modifiers.

The same goes for magic items that increase or modify ability scores, such as the tome of understanding (and other tomes), the manual of bodily health and the star card from the deck of many things, as well as items like gauntlets of ogre power increase scores instead of modifiers.

Reductions in abilities also works on scores: see strength drain of the shadow and devour intellect of the intellect devourer.

In the same way, scores are determined at character generation, rather than modifiers.

The strength score is additionally used for the calculation of carrying capacity and as prerequisite for some armor types.

Of course the scores will determine the modifiers as well. But modifiers are not changed directly.