Dwellers with high perception are best for the Water Treatment and Water Purification rooms. It also seems to help you pick locks and safes.

Perception raises the hit chance in combat and helps you find locations in the wasteland. Dwellers with high strength will also perform better in your Power Generator and Nuclear Generator. Some wasteland events also require strength like breaking open lockers and boxes. Strength is a stat that improves the damage dealt by your dweller, so high strength can useful for the defending of your vault, as well as exploring the wasteland.

First, I would like to explain what each stat does in this particular Fallout game and then on which stats you should focus in my opinion! If Fallout Shelter is your first Fallout game, the name SPECIAL is actually combination of the first letters of all stats a character in Fallout can have: Strength, Perception, Endurance, Charisma, Intelligence, Agility and Luck. Everyone familiar with the Fallout franchise has been confronted with the SPECIAL stats and in all games it has considerable influence on your character.

Fallout shelter exploring special