Active Fleet
Of Course I Still Love You
Autonomous Spaceport Droneship based on the US East Coast for offshore booster landings.
GO Ms. Tree
GO Ms. Tree is a fast, maneuverable vessel that has been specially modified to catch fairing halves.
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.
GO Ms. Chief
GO Ms. Chief is the near-identical sister ship to Ms. Tree and has been modified into a fairing catcher.
NRC Quest
NRC Quest is the sole Pacific Ocean support ship, recovering fairings. The ship previously recovered Cargo Dragon.
GO Quest
GO Quest is the droneship support ship. The vessel carries equipment and personnel to support booster landings at sea.
GO Searcher
GO Searcher is the primary Dragon recovery vessel, equipped with a medical treatment facility and heli-pad.

A Shortfall of Gravitas
Future Autonomous Spaceport Droneship that is assumed to be under construction based upon tweets by Elon Musk.
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.
Honourable Mentions

GO Discovery
GO Discovery is a ship occasionally used to transport Starship equipment between the development sites in Florida and Texas.

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

Hollywood
Former tugboat for OCISLY droneship throughout 2019. Later replaced by Tug Hawk, who was away for maintenance.
OCIRTI
Of Course I Read The Instructions. Starship spotting boat belonging to @SpacePadreIsle. Doggo included.