For a named player, find the section that starts with "04". Note: There is no space, colon, comma or anything between this 04 and the later numbers.
aka, find "|04". The numbers after that are the IDs of the ship types.
"|04239" is ship 239, the Lightning Frigate.
"|04239:230" is two ships, 239 - Lightning Frigate, and 230 - Spire Corvette.
Just keep adding ":ID" to the list.
For example, here's the important line from a recent game of mine:
22|003:0:12345:2:Toranth:-8888888:1|0123,2:123,2:123,2:123,2|04239:228:242:92|08888888880:80808:888888880|090:101:40:0:0|11False:False:0:0|120: <....>
The highlighted section is the part I'm referring to, that indicates which ships you have unlocked. Replacing ":92" with ":23" would drop Zenith Mirrors and add Parasites to my list of unlocked ships.
Here's a list of the ships and their ID numbers:
003 - Raider
008 - Space Tank
013 - Laser Gatling
016 - Sniper
020 - Armor
021 - Anti-Armor
022 - MLRS
023 - Parasite
024 - Raptor
027 - Space Plane
028 - ShieldBearer
030 - Vorticular Cutlass
031 - Attractor Drone
033 - Infiltrator
034 - SpiderBot
035 - Autocannon Minipod
036 - Teleport Raider
037 - Teleport Battlestation
038 - Bulletproof Fighter
039 - Electric Shuttle
045 - Tachyon MicroFighter
047 - Vampire Claw
048 - EyeBot
049 - Munitions Booster
080 - Zenith Bombard
081 - Zenith AutoBomb
082 - Grenade Launcher
089 - Zenith Electric Bomber
090 - Zenith Sentinel Frigate
092 - Zenith Mirror
093 - Zenith Paralyzer
094 - Armor Booster
095 - Zenith Beam Frigate
096 - Zenith Chameleon
097 - Zenith Polarizer
098 - Zenith Viral Shredder
099 - Impulse Reaction Emitter
100 - Acid Sprayer
130 - Neinzul Youngling Commando
131 - Neinzul Youngling Tiger
134 - Neinzul Youngling Vulture
135 - Neinzul Youngling Weasel
139 - Neinzul Youngling Nanoswarm
157 - Spire Armor Rotter
158 - Spire Blade Spawner
160 - Spire Gravity Drain
161 - Spire Gravity Ripper
162 - Spire Maw
163 - Spire MiniRam
164 - Spire Stealth Battleship
165 - Spire Teleporting Leech
166 - Spire Tractor Platform
204 - Zenith Medic Frigate
205 - Tackle Drone Launcher
210 - Neinzul Scapegoat
211 - Spire Railcluster
217 - Zenith Siege Engine
218 - Zenith Reprocessor
228 - Neinzul Youngling Firefly
229 - Saboteur
230 - Spire Corvette
239 - Lightning Frigate
240 - Zenith Devastator
241 - Neinzul Railpod
242 - Zenith Hydra
244 - Protector Starship
246 - Neinzul Youngling Shrike
245 - Neinzul Combat Carrier
250 - Powerslaver
251 - Lightning Starship
252 - Translocator Starship
Note: Adding ships doesn't remove them from the potential of an ARS trying to unlock it later, as that's stored in the planet data later in the save file. No idea what would happen in that case.