An easy way, which you may consider cheating, is using a map viewer .
Without doing that, there are at least two ways: Find mossy cobblestone, which is guaranteed to be beside a dungeon (see the first paragraph of the Minecraft Wiki article on dungeons ): Listen for a mass of zombie or spider sounds underground.
They love to make noise and you can hear them through walls, making them easy to find.
There's a chance that you'll find an underground cave or a dungeon near them.
Creepers do not make noise but there are no such creeper dungeons.
Some guesses:
Levels have relatively little pattern between components like caves and veins so dungeons might be completely randomly placed, even near the top of the levels: I'd guess that deeper down
you have a better chance, but without diving into the code there's no way to confirm this.
Out of interest: If your goal is to generate a map with lots of dungeons then Minecraft Seeds provides random seeds that do exactly that.
Here are three examples: Two nearby dungeons after climbing a mountain.
Three dungeons at spawn if you're too lazy to climb.
Inside a dungeon if you're too lazy to walk.
Note that an update to map generation could change what a seed produces.
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You can load an "Xray" texture pack that lets you look through most of the most common blocks, like dirt, gravel, and stone.
Dungeon rooms will standout when you look through the world as a rectangular room.
These are easy to find.
They may need to be repackaged as a .zip file.
However, this is out-and-out cheating by most accounts.
But this lets you see all sorts of amazing things...
and it's even compatible with SMP modes.
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I was walking around and I started hearing zombie sounds, so I quick started running and turned around, but I didn't see anything.
So I went closer, and I heard it again.
I was wondering if there was a dungeon
so I did this: If you press F on the keyboard, it changes the fog distance.
For a short time period, while it's changing, everything disappears except fire.
So if you press F rapidly, all you see is fire.
If there is a dungeon, the fire from a mob spawner looks different than torches and other kinds of fire, and it's cube-shaped, so it should be pretty obvious.
I tried this looking down, and sure enough, there was a dungeon right under me about 3 blocks down, near a beach.
It was a little ways from my spawn point, like maybe 100 blocks, but it suprized me because of how close to the surface it was.
I didn't dig into it at that time, because it was approaching nighttime, but I put a bunch of torches down, marking the spot.
That's one good way to spot dungeons.
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I found a dungeon about 50 blocks from my spawn point and about 15 blocks below the surface.
I've also heard of them appearing one or two blocks below the surface
so I don't think you need to dig deep.
I killed all the monsters and quickly planted one torch on the block next to each face of the spawn block and that stops anything from spawning.
(died a couple times)
I gathered all the mossy cobblestone since currently this is the only way to get it.
To find more of these my plan is to pick a direction and dig a main hallway with perpendicular branches going off on each side for at least 30 blocks.
My thinking is that this pattern will give me the best chance of finding anything in that general area as well as iron ore.
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A good way to tell there is a dungeon nearby is if you're wandering the map and find tunnels that are mostly flat and are fairly wide open.
(Maybe 3x3, 3x4, 4x4, 4x5 wide)
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In version 1.8, you can use customized world type to spawn more or less dungeons.
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In Minecraft survival mode, there are dungeon rooms with mob spawning machines, mossy cobblestones and special treasures.  What's the best way to find those?  I found one the other day and have been looking for more because it was great fun to beat, might be fun to farm, and had awesome special prizes.
How do I find Dungeons in Minecraft?
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<minecraft-java-edition>
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Look for tunnels that are mostly flat and wide open. You can also look for mossy cobblestones which are guaranteed to be beside a dungeon. Users are torn between loading external software in order to locate dungeons or spotting them in the game. Ways to spot them include looking out for flat tunnels, mossy cobblestone or listening for mass zombie sounds.
Users suggest loading external software. Users suggest ways to spot dungeons in the game.