Dungeon mob farming is pretty simple. You just have to find a Dungeon and make an automatic mob killer around the monster spawner. To find a dungeon you will have to search for one in caves, or use mods.
Preparation
One of the advantages of this method is that it requires very little preparation. You only need 3-4 signs, a couple of pressure plates, a couple of water buckets and a lava bucket. Plus the basic digging gear (Pickaxe, Shovel, etc.)
Finding the dungeon
This is the hardest part of the whole project. Dungeons are fairly rare and full of mobs. First of all, you can set the difficulty to peaceful, so that no hostile mobs appear. Then just start finding a dungeon. It is up to you how you find it, some people just dig around, some people use a map viewer such as cartograph (See Programs and Editors). Another easy way to find a Dungeon is by pressing F3 and a code will appear above mobs' heads. Hostile mobs usually have longer codes. If you see 3 or 4 codes in a small space, there is a large chance there may be a Dungeon there. Once you've found a dungeon, start building the killer. Make sure you know what the chosen Dungeon holds, because you may have to change the Lava Trap more suited to Spiders, as they can 'clog' the Lava Trap up.
Making the killer
First you have to take out all the valuable items inside chests. Make sure that the ceiling is 2 blocks high. Then just dig a 1x1 ditch around the spawner. Pick one corner of the ditch. Then do like this: 1= 1 block deep 2= 2 blocks deep X= 4 blocks deep
1
1
2
2
X 2 2 1 1
Then you should make a lava killer and a ditch that will make goodies go collection point.
A larger list for mob harvesters can be found here.
More Ways
Another way to make a dungeon trap is to go to the nether and collect 10-20 netherrack and make a flint and steel. About 3-5 blocks away make a square/circle around the spawner and another 3-5 blocks away make another square/circle with netherrack around the first circle/square. Now for the fun part, take your flint and steel and light all the netherrack on fire now when mobs spawn they will walk through the first fire trap and get set on fire, then the second one will finish them off, leaving their drops for you to enjoy!
For dungeons found close to the surface, you can make a trap that will allow mobs to spawn at night and burn during the day from the sun leaving all the goodies for you to collect in the morning without having to worry about anything spawning on you. All you have to do is dig up to the surface and place glass on top so that light will shine in.
The Lava Grinder
For skeletons and zombies, the way I made my mob trap is by clearing out a 11x11x4 area around the mob spawner. Mob spawners spawn in a 9x9 region centered around the spawner, so giving them that full region guarantees that they'll always be able to spawn, thereby maxing the spawn rate. Then in the borders, dig a canal and on the opposite end of the water source, dig the path to your collection area so you have something like this (top view):
xxxxxxxxxxxxx xwwwwwSwwwwwx xwbbbbbbbbbwx xwbbbbbbbbbwx xwbbbbbbbbbwx xwbbbbbbbbbwx xwbbbbMbbbbwx xwbbbbbbbbbwx xwbbbbbbbbbwx xwbbbbbbbbbwx xwbbbbbbbbbwx xwwwwwwwwwwwx xxxxxxwxxxxxx
S is water source
x is walls
w is flowing water
b is the floor around the spawner
M is the mob spawner
Then where the water flows out, you just make a simple 1 wide lava grinder like this (side view):
xxxxxxx-- --fffGx-- wwLxxxx-- xxSwwwwwx xxxxxxxxx
- is air
G is lava source
f is flowing lava
L is ladder (or a sign)
Then you just stand in the flowing water at the end and afk while your inventory fills up with feathers or bones and arrows.
Examples
Here[1] is a good example on how to make a dungeon trap using a lava grinder.