By official, I mean that the backgrounds are available for Adventurers League Play. While other backgrounds from Unearthed Arcana or from other WotC sources outside the D&D team are of interest, please mention whether they're considered official and sanctioned or not.
Current as of AL Player's Guide v10.3.
The Adventurers League Player's Guide covers what is a legal DDAL resource. From the AL Player's Guide v10.3:
Choosing Your Background. Choose a background from the PH or any fifth edition product listed above. Alternatively, you can instead customize a background using the rules found in the PH. Your choice of background is not restricted by the PH + 1 rule.
Eligible sourcebooks containing backgrounds:
- Basic Rules (BR), Chapter 3: Classes
- Player's Handbook (PHB), Chapter 4: Backgrounds
- AL Player's Guide v10.3, Appendix 2: Renown
Notably, Sword Coast Adventurer's Guide has been removed from the list of eligible sourcebooks, and the statement "Your choice of background is not restricted by the PH + 1 rule" is trivial as the PH is the only eligible source containing any backgrounds.
- Acolyte (BR, PHB)
- Charlatan (PHB)
- Criminal (BR, PHB)
- Criminal Variant: Spy (BR, PHB)
- Entertainer (PHB)
- Entertainer Variant: Gladiator (PHB)
- Folk Hero (BR, PHB)
- Guild Artisan (PHB)
- Guild Artisan Variant: Guild Merchant (PHB)
- Hermit (PHB)
- Noble (BR, PHB)
- Noble Variant: Knight (PHB)
- Outlander (PHB)
- Safe Haven (ALPG v10.3)1
- Sage (BR, PHB)
- Sailor (PHB)
- Sailor Variant: Pirate (PHB)
- Soldier (BR, PHB)
- Urchin (PHB)
Also worth noting is that D&D Adventurer's League supports customizing a background, as per the PHB rules for doing so. While this allows for a custom selection of skill proficiency and tool/language proficiency, you're still restricted to equipment packages and background features as published.
MOONSHAE ISLES & BORDER KINGDOM CHARACTERS
If you’re participating in adventures set in either of these locations, you can use their respective products (found on the DMs Guild) to select a background for your character. Rising Shadows-Moonshae Isles Regional Guide
The Border Kingdoms: A Forgotten Realms Campaign Supplement
1 The ALPG S10 FAQ states:
If backgrounds can be custom, does that mean there’s no restriction on what resource they come from?
You may create a custom background, but background features from the sources listed in the player’s guide are the only ones allowed. The only exception is the Safe Haven feature, the rules for which are included in the player’s guide.