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!]
Best Servants [★4]: 4 Star Tier List
Halloween Strike! Event Missions [JP Halloween 2017]
Best Craft Essence: Craft Essence Tier List
Cu Chulainn (Alter, Berserker)
Jeanne d’Arc (Alter)
Cheer for Master
Demon King of the Sixth Heaven
Talk on the Heated Sands