Active Fleet
Of Course I Still Love You
Autonomous Spaceport Droneship based on the US East Coast for offshore booster landings.

Shelia Bordelon
Shelia Bordelon is the new SpaceX fairing recovery vessel. Equipped with a 50 ton crane to lift fairing halves from the water.
Hawk
Hawk is a tugboat that is used to tow droneships to landing zones in the Atlantic Ocean.
Just Read the Instructions
Autonomous Spaceport Droneship based on the US East Coast for offshore booster landings.

A Shortfall of Gravitas
Future Autonomous Spaceport Droneship that is assumed to be under construction based upon tweets by Elon Musk.
NRC Quest
NRC Quest is the sole Pacific Ocean support ship, recovering fairings. The ship previously recovered Cargo Dragon.
Robots
Octagrabber
The Falcon 9 Securing Robot, known as Octagrabber, operates on both SpaceX droneships to secure boosters after a landing.
Spaceports

Phobos
Phobos is one of two Starship floating spaceports that SpaceX is currently constructing.

Deimos
Deimos is one of two Starship floating spaceports that SpaceX is currently constructing.
Former Fleet / Honourable Mentions
GO Ms. Tree
GO Ms. Tree is a fast, maneuverable vessel that has been specially modified to catch fairing halves.

GO Discovery
GO Discovery is a ship occasionally used to transport Starship equipment between the development sites in Florida and Texas.
GO Ms. Chief
GO Ms. Chief is the near-identical sister ship to Ms. Tree and has been modified into a fairing catcher.

GO Pursuit
A former fairing recovery ship based in Florida. Used throughout 2018 with varying success. Later replaced by Ms. Tree.

Lauren Foss
Lauren Foss is a tugboat that is occasionally to tow droneships to landing zones in the Atlantic Ocean.
OCIRTI
Of Course I Read The Instructions. Starship spotting boat belonging to @SpacePadreIsle. Doggo included.