What does each SPECIAL stat in Fallout Shelter do?

Do note that nothing has been made official, and this is a compilation of observations/hypotheses.

Strength

• Vault:
  • Decreased timer while working in power plants
• Wastes:
  • Needed to pry open lockers and boxes.
  • May also affect damage done by guns.
  • May increase a Dwellers capacity.

Perception

• Vault:
  • Hit chance in combat
  • Decreases water production timer of the Water Treatment and Water Purification rooms
• Wastes:
  • Find stuff like abandoned buildings
  • Critical attack cross hairs expand and contract slower, making it easier to deal maximum damage (×5) with a critical attack.

Endurance

• General:
  • Health points gained on level up (does not increase hp retroactively, max out endurance to 10+7 outfit and then start leveling to get maximum health)
  • Resist radiation and damage (11 Endurance = immune to radiation)
• Vault:
  • Food and water production timers of the Nuka-Cola bottler room.

Charisma

• Vault:
  • Improved fertility. (Reduced time to get pregnant.)
  • Decreases production timer in Radio room.
• Wastes:
  • Also chance of befriending people in the wastes (Gives the xp but no damage taken)
  • Interaction with groups of slaves

Intelligence

• Vault:
  • Med-bay (Stims and Radaway) production timers
• Wastes
  • Heal/help people

UPDATE:

• Vault/Wastelands:
  • also increases Exp gain (for level and Special stat increasing), extinguishing fire

Agility

• Vault:
  • Food production timer of the Diner and Garden rooms.
  • Increases attack speed.
• Wastes:
  • Run away from unwinnable fights taking less damage

Luck

• Vault:
  • Decreases chance of rush failure.
  • Increases amount of caps earned when resources collected from a production room (also increases chance to earn those caps)
  • Increases critical bar fill up rate.
• Wastes
  • Amount of loot found outside vault

Further reading and sources:

• Reddit: /r/foshelter What does each attribute do?
• Reddit: /r/foshelter Guys, power armor doesn't give any defensive stats in the wastes. ...
• Loading screen tips from Fallout Shelter for iOS

Tip:

• I set my shelter that way, to have the medbay spread in the vault to handle fire faster.
• the difference in having 1 or 2 intelligence is that a dweller with 2 intelligence levels up in half the time
• queuing my new matured dwellers to go to school so they get intelligence 2 or 3 then endurance for max HP and after that their main stat - super efficient

UPDATE: If you want to nerd out on more specifics, I recommend Section 7 of this Fallout Shelter Efficiency Data.

Also, here someone suggests that skills above 10 continue to benefit the dweller/explorer:

• Room efficiency data thread on Reddit

Effects during quests only:

• Strength: Increased attack damage
• Perception: Reduced critical hit speed
• Endurance: Reduced radiation damage
• Charisma: Better dialogue choices
• Intelligence: Increased "this is the right/wrong way" frequency
• Agility: Reduced time between attack rounds
• Luck: Increased chance of critical hit

These are the other uses for S.P.E.C.I.A.L. stats:

• Strength: Damage increased everywhere.
• Perception: Damage increased in-Vault; better Loot in the Wasteland, and better Critical lock-on on Quests.
• Endurance: Increased health.
• Charisma: Helps charming Wasteland Creatures. (And eventually better results in dialouges on Quests.)
• Intelligence: Increased medical capacities in the Wasteland.
• Agility: Better Damage in-Vault, better Hit Speed on Quests, and less Damage from losing a combat in the Wasteland.
• Luck: Also gives better Loot on Quests and in the Wasteland.