It's important to realize that mushroom biomes only spawn as islands in the middle of oceans.
So, if you want to look for one in a world that doesn't spawn you in one, you can't go walking to find one, you need to find an ocean, make a boat, and search the ocean for a mushroom island.
I can't offer any personal experience of how easy it is to find one by that approach
(I've no idea whether most oceans contain one, more than one, or just have a chance of containing one).
gaming.stackexchange.com/users/18830/Theodore Murdock
I'm afraid you have to get lucky.
Your best bet is to start a world with a seed that spawns in a mushroom biome, e.g. lost . ...or swim for a long time until you find one.
gaming.stackexchange.com/users/3610/fredley
Install the planes mod, build a p-51 with a radial engine, and have fun exploring almost faster than the map can load/generate.
This should make locating mushroom biomes a lot faster.
gaming.stackexchange.com/users/25489/Jakbruce2012
Well, I'm not bragging, I was really lucky.
In my Legit survival world I found a mushroom biome.
And here are some of my tips to find them: Scout the Ocean.
Make sure that the ocean is big and don't go next to the coast line, just keep the coastline visible in the corner of your eye.
Scout all coast lines of your map.
Since oceans are pretty much endless, you can walk on the beaches until you find one that is connected to your beach (worked for me) Use a seed with one either near, or in the map near spawn.
gaming.stackexchange.com/users/23949/alaskafish
Mobs cannot spawn on mycelium and the best way to find one is to make a boat and explore out in the middle of nowhere, kind of like finding diamonds.
gaming.stackexchange.com/users/25827/jose
The best "out of game" means I've found for identifying mushroom islands (and any other feature for that matter) is to use the third party tool AMIDST .
Before version 1.7.2 mushroom islands were extremely rare.
(I haven't explored much in 1.7.2 yet, to know how that has changed.)
gaming.stackexchange.com/users/16398/John
Not actually a biome location answer:
Mycelium spreads on dirt much like grass; if you're playing in SMP with the /give command (or other some other use of a inventory editor), you could give yourself Mycelium and simulate your own biome that way.
It wouldn't be a perfect recreation of Mushroom Biome (if you built over a Desert biome, you may not be able to raise huge mushrooms); Mycelium only allows spawning of Mooshrooms, whereas the Biome itself is what causes anti-mob and natural giant mushrooms.
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gaming.stackexchange.com/users/14438/Broam
Minecraft's mushroom biome sounds like a pleasant place: Food in abundant supply, and zero hostile mob spawns.  However, in all my time playing, I've never ever found one. Is there a trick to finding them, or do you just have to get lucky?
How do I find mushroom biomes?
gaming.stackexchange.com
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gaming.stackexchange.com/questions/45563
gaming.stackexchange.com/users/9752/a cat
Make a boat and scout the ocean for a mushroom island.  Detailed explanation as to how to find the location in question, is offered. Also, there are different recommendations, regarding that matter, such as third party softwares.
The community points the location. Usage of a third party software is suggested.