TRUE EB0LA
Exotic Weapon Catalyst

Notes:
1 PVP kill counts for around 20 PVE kills (this number could be wrong, but that is from what I have seen so far)
Crucible Victories doesn't require an actual victory, just kills will suffice.
The only requirement to get the catalyst to drop is to own the exotic. You do not need it equipped or even on your character. (Worldline Zero is special as it requires sword kills (even with no ammo) to get the catalyst to drop.
Faction Rally Catalysts will be able to be earned later in another way
Borealis
Catalyst: +20 Reload Speed
Where to get it: any PVE enemy
How to Unlock:
150 Arc Kills
150 Solar Kills
150 Void Kills
Node Available: Yes
Coldheart
Catalyst: +20 Stability / +20 Reload Speed
Where to get it: Heroic Strike/Nightfall bosses
How to Unlock: 300 kills
Node Available: Yes
The Colony
Catalyst: +10 Magazine / +30 Weapon Size / Deeper Pockets (Perk) Increased ammo reserves.
Where to get it: Crucible victories
How to Unlock: 300 kills
Node Available: Yes
Crimson
Catalyst: +20 Range
Where to get it: any PvE kill
How to Unlock: 300 precision kills
Node Available: Yes
D.A.R.C.I.
Catalyst: +20 Stability
Where to get it: any PvE kill
How to Unlock: 300 precision kills
Node Available: Yes
Fighting Lion
Catalyst: +30 Reload Speed / Chimera (Perk) After this weapon fires, equipped Kinetic and Power weapons gain a brief period of increased handling and precision damage.
Where to get it: Unknown
How to Unlock: Unkown
Node Available: No
Graviton Lance
Catalyst: +20 Range / +15 Aim Assist / Hidden Hand (Perk) Better target acquisition.
Where to get it: Achieve Rank 50 Dead Orbit
How to Unlock: 500 kills and 1000 kills with Cosmology perk
Node Available: Yes
Hard Light
Catalyst: +25 Stability
Where to get it: Found by engaging in activities during the celebration of deadly craftsmanship. (Armsweek)
How to Unlock: Defeat 1,000 enemies and Complete the Nightfall strike "Savathûn's Song" during Armsweek with Hard Light equipped.
Node Available: No
The Huckleberry
Catalyst: Ambitious Assassin (Perk) Overflows the magazine based on the number of rapid kills before reloading.
Where to get it: Heroic Adventures
How to Unlock: 300 kills
Node Available: Yes
Jade Rabbit
Catalyst: +30 Stability
Where to get it: Crucible victories
How to Unlock: 250 kills in crucible
Node Available: Yes
Legend of Acrius
Catalyst: +40 Reload Speed/+40 Magazine size/Deeper Pockets perk
Where to get it: Found in the greatest challenges aboard the emperor's craft. (Prestige Leviathan)
How to Unlock: 300 TBD
Node Available: No
Merciless
Catalyst: +40 Range / +40 Stability
Where to get it: Heroic Strike/Nightfall bosses
How to Unlock: 500 kills
Node Available: Yes
MIDA Multi-Tool
Catalyst: Outlaw (Perk) Precision kills greatly decrease reload time.
Where to get it: Competitive PVP victories
How to Unlock: 250 precision kills in Crucible
Node Available: Yes
Polaris Lance
Catalyst: Dragonfly (Perk) Precision kills create an elemental damage explosion.
Where to get it: Found by pursuing the secrets of a newly awakened intelligence
How to Unlock: 500 kills and 50 kills with Perfect shot perk
Node Available: Yes
Prometheus Lens
Catalyst: +20 Stability / +20 Handling
Where to get it: Heroic Strike/Nightfall bosses
How to Unlock: 300 kills
Node Available: Yes
The Prospector
Catalyst: +40 Blast Radius / +70 Weapon Size / Deeper Pockets (Perk) Increased ammo reserves.
Where to get it: Found by engaging in activities during the celebration of deadly craftsmanship. (Armsweek)
How to Unlock: Defeat 500 enemies and complete the Nightfall strike "The Arms Dealer" during Armsweek with the Prospector equipped.
Node Available: No
Rat King
Catalyst: +20 Aim Assistance / +20 Recoil Direction / Infestation (Perk) Improved aim assist and recoil direction. Refreshes health on Vermin invisibility trigger.
Where to get it: Found by engaging in activities during the celebration of deadly craftsmanship. (Armsweek)
How to Unlock: Defeat 1,000 enemies using Rat King to unlock this upgrade. (Also shows a Nightfall Completion is required but does not specify which)
Node Available: No
Riskrunner
Catalyst: +30 Range
Where to get it: Heroic Strike/Nightfall bosses
How to Unlock: 300 kills and 300 kills with chain lightning
Node Available: Yes
Skyburner's Oath
Catalyst: +30 Range
Where to get it: Leviathan Raid completion
How to Unlock: 1000 Cabal kills
Node Available: Yes
Sleeper Simulant
Catalyst: +75 Charge Time / Accelerated Coils (Perk) Speeds up weapon charge.
Where to get it: Found in the greatest challenges of the astral pinnacle. (Prestige Spire of Stars)
How to Unlock: 500 TBD
Node Available: No
Sturm
Catalyst: +20 Range / +40 Handling
Where to get it: Unknown
How to Unlock: Unkown
Node Available: No
Sunshot
Catalyst: +30 Range / +20 Stability
Where to get it: Achieve Rank 50 Future War Cult
How to Unlock: 500 kills and 1000 kills with Sun Blast Perk
Node Available: Yes
SUROS Regime
Catalyst: +50 Recoil Direction / SUROS Ascendance (Perk) Increased chance for kills to regen health.
Where to get it: Found by engaging in activities during the celebration of deadly craftsmanship. (Armsweek)
How to Unlock: Defeat 300 Crucible Enemies and complete Strange Terrain Nightfall during Armsweek.
Node Available: No
Sweet Business
Catalyst: Serious Business (Perk) When this weapon is fully spun up, the flinch from incoming damage is greatly reduced.
Where to get it: Achieve Rank 50 New Monarchy
How to Unlock: 250 multi-kills (4 kills in a row)
Node Available: Yes
Telesto
Catalyst: +60 Magazine / +40 Weapon Size / Deeper Pockets (Perk) Increased ammo reserves.
Where to get it: Found in the greatest challenges of the planet devourer. (Prestige Eater of Worlds)
How to Unlock: 500 TBD
Node Available: No
Tractor Cannon
Catalyst: +70 Magazine / +50 Weapon Size / Deeper Pockets (Perk) Increased ammo reserves.
Where to get it: any PvE kill
How to Unlock: 300 kills
Node Available: Yes
Vigilance Wing
Catalyst: Full Auto Trigger System (Perk) Holding down the trigger will fire this weapon at full auto.
Where to get it: Crucible victories
How to Unlock: 250 PVP kills/5 Blood for Blood medals (avenge a teammate)
Node Available: Yes
Wardcliff Coil
Catalyst: Pinpoint Guidance Module (Perk) Improved tracking on projectiles.
Where to get it: Heroic Strike/Nightfall bosses
How to Unlock: 500 kills
Node Available: Yes
Wordline Zero
Catalyst: Another Dimension (Perk) Reduces the activation time of Tesseract.
Where to get it: any PvE kill with a sword
How to Unlock: Hit each Level 7 EP boss (5 in total) with Worldline Zero (unshielded)
Node Available: Yes
Videos:
A. How to unlock Graviton Lance
B. How to unlock Polaris Lance
C. How to unlock Sunshot (see Sweet Business video for location)
D. How to unlock Sweet Business