Earth Chest 01
Objective: Find a warp drive
After accessing the cosmodrome network (interacting with the terminal with your ghost), head outside and follow the marker up some stairs into a large structure. The chest will be on a desk to the left inside the second room.
Earth Chest 02
Mission: The Dark Within
Objective: Kill the hive wizard
After leaving the darkness area filled with hive acolytes and a knight. Head up the stairs to reach a room with thralls, acolytes and a wizard. Kill all the enemies except the wizard (killing it ends the mission) and leap up onto the top of what looks like a small office space (the roof area you should land on is made of corrugated iron). You should see the chest in the far back corner of this small area.
Earth Chest 03
Mission: The Warmind
Objective: Reach the forgotten shore
Follow your waypoint marker from the start of the mission until you reach the long canyon area leading to the forgotten shore. Partway along is a set of small waterfalls, underneath one is a small cave which contains the chest.
Earth Chest 04
Mission: The Warmind
Objective: Infiltrate the skywatch
From the amplifier on the ship (the one you just accessed in the prior objective), look towards the nearby cliff (away from the second amplifier location) with the shallow water below it. Drop down and walk into the shallow water in the small inlet and walk along the base of the cliff path. If you keep jumping as you go, on one of the shelves of rock poking out from the cliff, you will find the chest. If you get the turn back message, you've gone too far.
Earth Chest 05
Mission: The Devil's Lair (strike)
Objective: Destroy sepiks prime
This can be done during or after the final boss battle in the strike mission. In the final room of the area there are two height levels. Drop down underneath the platform you start on and look on the right side against the wall (right when facing towards where the boss is/was). The chest should be easily visible.
Moon Chest 01
This can be done during a mission but it is quicker to just do it in patrol. From your starting point, head up the slope and left. Go past the structure on your sparrow and into the small canyon beyond. Just past the entrance of the canyon you will see a ramp to your right with a large crevasse underneath it (the first picture above). To your left from the ramp is a rocky outcrop. Among the stones is the golden chest.
Moon Chest 02
Mission: The World's Grave
Objective: Enter the hive fortress
As you move forward in the hive fortress, you will come to a room with what looks like large white maggots or cocoons on the walls that has a path circling around a central pillar multiple times. At the bottom of this is a large pool of water. Walk through the water and hidden in an alcove is the chest.
Moon Chest 03
Mission: The World's Grave
Objective: Push back the hive
In the final room of the World's Grave mission, with several knights kneeling around a pillar. Head for the pool of water on the left when you enter and you should find it sitting on a shelf of rock.
Moon Chest 04
You need to first head to the anchor of light (take the right path from your spawning location and follow the trail). When you reach the anchor (a large circular building sitting on a narrow pillar), turn left through a small gap in the canyon walls to find the trail to the shrine of crota. Head into the shrine and down a couple of floors until you enter a circular room with a large drop surrounding a platform in the middle. After eliminating the enemies here, look down to see a set of platforms attached to the wall partway down the drop. These can be used to reach the chest.
Moon Chest 05
Mission: The Summoning Pits (strike)
Objective: Defeat Phogoth
In the final part of the strike when you enter Phogoth's chamber, immediately turn left and head for the platform at the far end (it is slightly lower than the rest of the area) where the hive ships arrive from. On this platform is the final moon chest.
Venus Chest 01
From your starting location, take the right path towards the academy area of the planet. Push past the first group of fallen enemies and when you reach the second, turn left and enter the nearby doorway at the top of the stairs. Just inside is a hole with glowing light pouring out of it. Head inside and look behind one of the pillars on the right to find the chest.
Venus Chest 02
The next chest is in the N/Gen area of the academy. From your starting location, take the right path towards the academy area of the planet. Push past the first group of fallen enemies and when you reach the second, turn left and enter the nearby doorway at the top of the stairs. Just inside is a hole with glowing light pouring out of it. Head inside and out the other end to reach the Ishtar Academy area. From here pass to the far end of the hall to reach the Ishtar Commons. The doorway to the N/Gen area has two hobgoblins guarding it, along with a full scale vex and fallen battle going on nearby. Enter the N/Gen area and look for a detached structure to the right. Inside this is a set of cubicles. Inside one of them is the chest.
Venus Chest 03
This chest is in the Campus 9 area. To reach it, take the right path from your starting point and keep following the trail (from the shattered coast to near the end of the waking ruins area). Keep following the trail when the area opens up in the waking ruins until you see a huge circular hole in a piece of stone. Enter the hole and follow the short path to reach the Campus 9 area. When you arrive, look for an orange building at the back of the area. Behind the building in a small alcove is the chest.
Venus Chest 04
Mission: Scourge of Winter
Objective: Find Draksis
Fight your way through the ember caves area (the cave network area of the mission) In the area just after you obtain your new objective (before you reach the outside air once again) is a large hole in the ground with the chest visible on a rock ledge opposite. Jump across the gap to open it.
Venus Chest 05
Mission: Scourge of Winter
Objective: Storm the Ketch
On the deck of the Kell's ship (after killing the two sniper vandals) look to the right and head for the rear of the ship (away from the opening to head inside). Right at the stern is the chest.
Mars Chest 01
From your starting position you should nearly be able to see the bunker where the chest is. Head towards the cabal forces straight ahead and look for the dome shaped building popping up out of the sand. Inside on a box is the chest.
Mars Chest 02
From your starting point, take the left trail until you go through a tunnel dividing the starting area and the next one. Near the far end of the tunnel is a hole in the roof that can be entered to find the chest.
Mars Chest 03
This one is inside the Dust Palace area (part of the sunken city). Look for the large rounded Clovis Bay building (in blue paint) and head inside. Work your way down to the first large open area inside and look for a white number three with a faded five next to one of the doors in this area. Inside, hidden behind a desk is the chest.
Mars Chest 04
Mission: Cerberus Vae III (strike)
Objective: Find Valus Ta'aurc
Shortly after starting the strike mission, you will enter a tunnel section (the Iron Line area). A tunnel door will open and across from you will be a large circular door with green light illuminating it. Above and to the right of this door is where the chest is hidden, To reach it you need to jump on the pipes to the left of the large tunnel door and then look up to see a platform nearby to your right. Jump up there and then across to the adjacent platform to reach the chest.
Mars Chest 05
Mission: A Rising Tide
Objective: Destroy the gates
Near the end of the mission, you will enter an area where vex spawn out of transport gates. Look for a gate with a large white two on the wall to the right of it and enter the doorway next to the white two. Inside will be a wire fence, behind which is hiding the final chest.