Here is the full list of all 29 Lovers in a Dangerous Spacetime trophies - there are 15 bronze, 6 silver, 7 gold and 1 platinum.
Collect all other trophies.
Beat any level in two player mode.
Defeat King Cepheus.
Remove all of King Cepheus's shields.
Rescue every captured friend.
Kill 20 enemies with security guns in a single level.
Make two Power-Metal missiles collide.
Escape a white dwarf without shooting or ramming it.
Play through a campaign without using any gems.
Earn one of each type of ship badge.
Survive after being fully enveloped by a bomb trap.
Defeat Orion.
Deliver the final blow to Orion using a solar flare.
Fill up all gem slots in a fully upgraded ship.
Have 20 cells at one time on the Metal-Beam shield.
Defeat Cetus while its head is underwater.
Finish a level without crossing paths with your partner.
Defeat Cetus.
Use three different types of Yamato weapons in a boss fight.
Finish a level without taking damage.
Defeat Ursa Major without taking a punch.
Defeat Ursa Major.
Pirouette 15 times in a single level.
Throw a gem into a station that is being used by another player.
Have 10 unused gems in your ship at one time.
Use every station in a single level.
Finish a level after removing all fog of war.
Beat any level with the help of your trusty space-pet.
Finish the tutorial.