America: Charlotte (Free Quest) is the best AP-saving & time-saving location to farm Void's Dust. You'll get around 6x Void's Dust by spending 200 AP.
Fuyuki (Singularity F): Unknown Coordinates (X-F) is the first location you unlock to farm Void's Dust.
Shadow Servants drops Void's Dust. So you can get Void's Dust from Daily Quests as well.
Here is a list of servants that need Void's Dust to perform Ascension and Skill Up.
Rashoumon event (Ibaraki Douji event)
Murder at the Kogetsukan
Inheritance of Glory
5 Star Servants Tier List: Best Servants [5/9 updated!]
FGO Reroll Tier List [5/11 updated!]
1-3 Star Servants Tier List
Best Servants [★4]: 4 Star Tier List
Best Craft Essence: Craft Essence Tier List
Cu Chulainn (Alter, Berserker)
Nero Claudius (Caster, Summer 2017)
Minamoto No Yorimitsu (Lancer, Summer 2017)
Oda Nobunaga (Berserker, Summer 2017)
Hans Christian Andersen
Frankenstein (Saber, Summer 2017)
Sometime in Summer
Piece of Year 2030 (A Fragment of 2030)