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.
Finish the tutorial.
Beat any level in two player mode.
Beat any level with the help of your trusty space-pet.
Defeat Ursa Major.
Defeat King Cepheus.
Finish a level after removing all fog of war.
Finish a level without taking damage.
Finish a level without crossing paths with your partner.
Fill up all gem slots in a fully upgraded ship.
Play through a campaign without using any gems.
Earn one of each type of ship badge.
Rescue every captured friend.
Defeat Ursa Major without taking a punch.
Defeat Cetus while its head is underwater.
Deliver the final blow to Orion using a solar flare.
Remove all of King Cepheus's shields.
Use every station in a single level.
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 three different types of Yamato weapons in a boss fight.
Escape a white dwarf without shooting or ramming it.
Survive after being fully enveloped by a bomb trap.
Kill 20 enemies with security guns in a single level.
Have 20 cells at one time on the Metal-Beam shield.
Make two Power-Metal missiles collide.