All of the free ships in Starfield are unlockable through quests.
Kepler R/Kepler S
These ships are obtainable through the quest Overdesigned. You can only obtain one of these. The choices you make during the quest determine which one you get.
You get the quest by talking to Walter Stroud after the Constellation quest “All That Money Can Buy”.
He will ask you to visit his company’s space station in the Narion system and help his team design a new ship.
You’ll be asked about budget options; go for the biggest possible budget. You’ll need to pass a persuasion check to convince the team to go for it, but you won’t be able to get the Kepler R without it. Next, you will need to complete a few bounties from the nearby job board in order to get data on ship performance. Grab a passenger and a bounty mission to cross off more boxes from the objective list. These bounties are randomly generated, but they are very simple and easy to complete. After that you’ll have to go around the team and hear all of their big pitches for features on the ship. When the team lead asks you what to do to resolve this deadlock suggest that everyone makes one cut. Take the design back to Walter and you’ll receive Kepler R.
If you want the Kepler S you want to do the opposite, however I would not recommend it as Kepler R is better in every way.
Razorleaf
The quest Mantis, where this ship is from, is obtainable through reading “Secret Outpost!” Data slate which drop from dead Spacers. Additionally you also get a legendary space suit and helmet from this quest.
The slate will lead you to Denebola I-b, you need the slate in order to be able to land at the Mantis lair, otherwise it won’t show up on the moon as a landing option. The enemies you’ll encounter range from Level 8 to Level 30. Once you land you’ll be met with a couple of Spacers, kill them and proceed into the Lair. Go through the Lair and clear it, at the end of it there is a puzzle room with letters on the ground and turrets which will shoot you if you don’t proceed correctly.
The word you want to spell is Tyrannis.
Once you get past the turrets you’ll find both the ship Razorleaf and the legendary space suit. In order to board and fly the ship away you need to press a button that lifts the platform with the ship outside.
Star Eagle
The Star Eagle is acquirable by finishing the Freestar Collective questline which you can start after first arriving in Akila City and helping out with the bank robbery. After that you’ll be told to speak to a woman if you want to become a Freestar Collective Ranger. You start the questline by talking to her.
UC Prison Shuttle
You get the ship at the end of “Echoes of the Past” which is part of the Crimson Fleet questline. Although I can’t imagine why anyone would want this ship.
Wanderwell
This ship is one of the gifts you’ll eventually receive from your parents if you’ve chosen the “Kid Stuff” trait. It has its own questline with your parents, you’ll receive this ship at the end of it.
Spoilers
Starborn Guardian
You unlock this ship by completing the game and starting a New Game+ playthrough.
