I purchased Sword Coast Adventurer's Guide and none of the classes are appearing for me to use

I purchased Sword Coast Adventurer's Guide and now the classes I wanted to use are not in my options

Do you mean subclasses? (That book does not contain any new classes.)

If so, there are two main things to look into:

1. Go here and see if it says your entitlements are synced. If it says they're not, click the button there to sync them.

2. This book was written for the 5e rules. Many of the subclasses in it have been superseded in the 5.5e rules by updated versions. In these cases, if you've selected the 5.5e version of a class, you won't be able to use the 5e version of the subclass, and if you don't own the 5.5e content that includes it, you won't be able to use the 5.5e version either. In that case, you'd need to remove the 5.5e class and choose the 5e class instead; then you'll be able to select the 5e version of the subclass.

Specifically:

The following are still eligible for use in the 5.5e ruleset
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Barbarian - Battlerager
Cleric - Arcana Domain
Monk - Long Death, Sun Soul
Paladin - Oath of the Crown
Rogue - Mastermind, Swashbuckler
Sorcerer - Storm Sorcery
Warlock - Undying

The following have been replaced by 5.5e versions of the subclasses
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Battlerager - Totem Warrior (options provided, replaced by Wild Heart)
Fighter - Purple Dragon Knight (replaced by Banneret)
Wizard - Bladesinger

Also make sure the 5e Expanded toggle is turned on in your character settings.