I have not run 5e as of yet. However, Darksun or an Asian themed campaign aside, the following are most likely allowed
Races
All Campaigns: Human (variant)
Most Campaigns: Dwarf (Hill, Mountain (Mountains lose Heavy Armor Proficiency)), Half-Elf (non-drow), Elf (High, Wood), Gnome, Halfling, Lizardmen
Some campaigns: Goblin, Gnoll, Half-Orc, Half-Ogre, Kobold, Minotaur, Ogre, Orc,
Classes
Barbarian: Berserker (may drop for a Berserker type Fighter Archetype)
Bard: Lore, Valor
Cleric: may drop for a third party priest or divine warlock variant
Druid: Circle of Land
Fighter: Battlemaster, Commander (JesterDavid), Skald (I am not a fruit)
Fighter variant: Light Armor Fighter (Khaalis)
Paladin: Devotion, Ancients
Ranger variant: Spellless (Unearthed Arcana)
Rogue: Thief, Minstrel (I am not a fruit)
Sorcerer: Fey (JesterDavid)
Warlock: ArchFey (Great Old Ones are NPC (limited to certain campaigns), Fiend is NPC Only)
Wizard
Witch (a third party),
Witch (BMcDaniel)