Dark Souls 3

Scaling

Scaling determines how much additional damage a weapon deals.

The additional damage is based on the weapon's stat bonus, base damage and your strength, dexterity, faith and intelligence stats.
The weapon's stat bonus is indicated by a letter grade of S/A/B/C/D/E or as - which means no scaling for that stat.

Calculating bonus damage

Each weapon has a base bonus value for each stat, for example the Butcher Knife has a strength bonus of 104
This bonus gets multiplied by a scalar value dependent on the upgrade level. For any weapon maximally upgraded this scalar value is 1.4.
The result of this multiplication is divided by 100 and then multiplied by a value dependent on your strength stat. This gets multiplied by the weapon damage.
This calculation is done for every stat the weapon has a bonus for and then these values get added together to form the bonus damage.

Example
My character uses a Butcher Knife+10 with a strength bonus of 104 and 190 weapon damage.
It is a +10 weapon so the scalar value is 1.4.
My character has 25 strength, at 25 strength the stat's modifier is 0.4342289734.

Bonusdamage=1041.4/1000.4342289734190

This gives us a bonus damage of ~120.13 or 120.

*I haven't tested with intelligence or faith*

Weapon upgrade bonus scalars

Upgrade Type+0+1+2+3+4+5+6+7+8+9+10
Titanite Shard1.01.051.11.151.21.231.261.291.321.351.4
Twinkling Titanite1.01.081.161.241.321.4-----

Strength/Dex Modifiers

LevelScalar
10.0
180.25
400.75
600.85
991.0

Weapon Base Bonuses