Dexterity, Constitution, and Wisdom can be added permanently to AC. Intelligence can be added for a minute. Charisma can be added through a magic item from Hoard of the Dragon Queen or to a single attack. I found no way at all to add Strength to AC.
Below I will provide "the best" methods of adding a given ability modifier to one's Armor Class. By "best" I mean the one that lasts the longest, which, in the case of Charisma, is quite short. This means that features which add Constitution to AC for 1 attack, turn, round, minute, or anything else will not be mentioned/listed:
While you are not wearing any armor, your Armor Class equals 10 + your Dexterity modifier + your Constitution modifier. You can use a shield and still gain this benefit.
- Player's Handbook (page 48)
Beginning at 1st level, while you are wearing no armor and not wielding a shield, your AC equals 10 + your Dexterity modifier + your Wisdom modifier.
- Player's Handbook (page 78)
Bladesinging Wizard's Bladesong feature (Int):
[...] You can use a bonus action to start the Bladesong, which lasts for 1 minute. It ends early if you are incapacitated, if you don medium or heavy armor or a shield, or if you use two hands to make an attack with a weapon. You can also dismiss Bladesong at any time you choose (no action required).
While your bladesong is active, you gain the following benefits:
- You gain a bonus to your AC equal to your Intelligence modifier (minimum of +1) [...]
- Sword Coast Adventurer's Guide (page 143)
[...] When a creature you can see hits you with an attack roll, you can use your reaction to gain a bonus to AC against that attack, potentially causing it to miss you. The bonus equals your Charisma modifier (minimum of +1) [...]
- Bard and Paladin Unearthed Arcana (page 2) [link to document]
An item from Hoard of the Dragon Queen (Cha):
This item is the Black Dragon Mask which has the following property:
Draconic Mastery. While you are wearing no armor, you can add your Charisma bonus to your Armor Class.
Thanks to user David Coffron for pointing out this item!