Here is the full list of all 28 Lust Vessel achievements.
Reach 1 lust
Reach 50 lust
Reach 100 lust
After having diner, nobody wanted to go with Kate
After having diner, Kate convinced Erika to explore the ship
After having diner, Kate convinced Bob and Lucila to explore the ship
Kate had a nice diner with her new group of friends
Kate and Cheba met again
Kate rescued the girl from the posters
Kate met a mad scientist
Kate decided to escape the ship
Kate decided to stay and help solving the ship's problems
Kate is having too much fun, there is no reason to leave the ship
After escaping Kate went back to her "normal" life
After escaping Kate formed a relationship with Erika
After escaping the ship Myra quit her job to stay with Kate
After escaping the ship, Kate started working with Myra
Kate got a cat and a star leaving in the same house
After escaping the ship, Kate finally accepted Cheba's proposals
After escaping the ship, Kate got a normal boyfriend
Kate and Cheba got a boyfriend... and they are dating each other too
After escaping the ship, Kate got a husband
After escaping the ship, Kate got a wife
Kate managed to get a relationship, with a married couple
Kate saved the ship and managed to seduce a scientist
After saving the ship, Kate got to take home her own slimegirl
Kate loved her science work, so she took the whole lab back home
You saw the two parts of Kate's harem