This Destiny 2 The Sludge Lost Sectors page will explain where to find all the mini-dungeons within that area of the EDZ / Earth Patrol space.
If you want more information on how to find Lost Sectors and the rewards they offer, our Lost Sectors locations page can help.
Destiny 2 The Sludge Lost Sectors
The three Lost Sectors in The Sludge are:
Hallowed Grove location
Location: To find this symbol, start at The Sludge spawn point, turn right, head past the entrance to Shaft 13, following the wall to your right until you reach a small hill with the symbol painted on its front. The entrance to this dungeon is a sunken cave opening behind the hill. Enter it and turn right, then left.
Strategy: Follow the dimly lit tunnel as it curves to the right, the jump up onto the higher platform, then turn left. You'll now encounter your first wave of Taken, including a couple of Centurions, Psions and Hobgoblins. Focus fire on the Psions (as they can clone themselves) before letting loose on the rest.
Turn right and jump up on the raised platform. Focus fire on the Phalanx's and the Centurions (as they can inflict serious damage), then destroy the Psions to avoid getting mobbed. Also, watch out for a Hobgoblin sniping you on a raised platform to your left. Turn right to enter the final area.
The final chamber of this section is an open area filled with broken tree boughs. The boss, Vendraxis, is a tough cookie so you'll need to use you jump to good effect here.
Take out most of the Psions, and the Hobgoblins, and start hitting the boss with small bursts of fire (hand cannons and pulse rifles seem to work really well against him). He will teleport every few seconds and has a nasty shadow bomb that makes it difficult to see him. Use your double jump evade them and wear him down.
Shaft 13 location
Location: To find the symbol for the Lost Sector, turn right from The Sludge spawn point and follow the wall to your right.
You'll come to an open container and an open red set of doors. The symbol is to the left of these doors. Beware a group of Red Cabal Legionnaires patrolling this area.
Strategy: Pass through the red doors, and jump up onto the containers to your left. Head inside the door with the lit sign above it, turn left, turn right. Go down some stairs, turn right, down some more stairs, then turn right again. You'll now be in a semi-open area with blue light and an overturned truck.
Turn right to engage your first group of Fallen. These are just Vandals and Dregs, but make sure you have a Power Level of around 180-200, otherwise these guys alone will be a hard fight. Be wary of Vandals sniping you on a platform to your right.
Once these guys have been cleared out, move forwards and you'll encounter more Vandals and Dregs, as well as the boss of this dungeon.
Take out most of the Vandals first (we also encountered one Marauder, sneaking about in the structure next to where the boss is. As a Fallen captain, Calzar teleports a lot, especially when his shield is down. Use your super, and grenades to pummel him into a corner. The chest is on the top level above him.
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Cavern of Souls location
Location: This symbol can be found in the middle of The Sludge. Look for a warehouse type building on a hill. This area is crawling with Red Cabal, so take out the Phalanx's, Legionnaires and Psions before heading to direction show on the map (see screenshot). The symbol will be above the door.
Strategy: Walk under the symbol, and turn left, walking through the open container. Turn right and follow the tunnel down. It'll get really dark here. Turn right, head forwards, turn right, then turn left.
Now you'll encounter your first group of enemies, including Taken Psions and Centurions. Again, keep your fire focused on the Psions as their cloning ability can leave you mobbed.
Now turn right and look up. While this area is really dark, you should be able to see a Wizard floating about above you. Jump up and engage him.
There are plenty of Psions here, but the Wizard can summon more so focus you fire on him. Use your double jump to evade and take down the remaining Psions.
Turn right, then enter the tunnel in front of you, turn right again then jump up onto the raised platform. Turn right twice more and you'll be able to see the final chamber of this Lost Sector (it'll have a giant, static-coloured ball at the top of it).
The boss, Varghul, will be at the bottom along with plenty of Psions, Goblins and a few Vandals (sniping you from the left-hand side). Take down the Vandals then use cover to bring Varghul down - he has no shield so he shouldn't be too much of a risk.