Sources where they first became available to players:
PHB = Player’s Handbook
MM = Monsters Manual
MToF = Mordenkainen’s Tome of Foes
SCAG = Swords Coast Adventurer’s Guide
VGtM = Volo’s Guide to Monsters
GGtR = Guildmasters’ Guide to Ravnica
WGtE = Wayfinder’s Guide to Eberron
DMG = Dungeon Master’s Guide
Standard Languages:

Common (PHB)
Dwarvish (PHB)
Elvish (PHB)
Giant (PHB)
Gnomish (PHB)
Goblin (PHB)
Halfling (PHB)
Orc (PHB)

Standard Languages (Ravnica):

Abyssal (PHB)
Celestial (PHB)
Common (PHB)
Draconic (PHB)
Elvish (PHB)
Giant (PHB)
Goblin (PHB)
Kraul (GGtR)
Loxodon (GGtR)
Merfolk (GGtR)
Minotaur (GGtR)
Sphinx (MM)
Sylvan (PHB)
Vedalken (GGtR)

Exotic Languages:

Abyssal (PHB)
Celestial (PHB)
Draconic (PHB)
Deep Speech (PHB)
Infernal (PHB)
Primodial (PHB)
Aquan
Auran
Ignan
Terran
Sylvan (PHB)
Undercommon (PHB)

Race/Class-Specific Languages:

Aarakocra (MM)
Druidic (PHB)
Gith (MToF)
Thieves’ Cant (PHB)

Forgotten Realms Human Languages (requires GM permission):

(All in SCAG)

Dambrathan
Bedine
Alzhedo
Chondathan
Damaran
Waelan
Guran
Halruaan
Illuskan
Roushoum
Chessentan
Mulhorandi
Untheric
Thayan
Rashemi
Shaaran
Shou
Tuigan
Turmic
Uluik

Eberron’s Common Languages

Common (PHB)
Dwarvish (PHB)
Elvish (PHB)
Giant (PHB)
Goblin (PHB)
Orc (dead language, PHB)
Infernal (PHB)
Riedran (WGtE)
Quori (WGtE)

Monstrous Languages (available via Favored Enemy or Training):

Blink Dog (MM)
Bullywug (MM)
Giant Eagle (MM)
Giant Elk (MM)
Giant Owl (MM)
Gnoll (MM)
Grell (MM)
Grung (VGtM)
Hook Horror (MM)
Kruthik (MToF)
Modron (MM)
Otyugh (MM)
Sahuagin (MM)
Slaad (MM)
Sphinx (MM)
Thri-kreen (MM)
Tlincalli (VGtM)
Troglodyte (MM)
Umber Hulk (MM)
Vegepygmy (VGtM)
Winter Wolf (MM)
Worg (MM)
Yeti (MM)