From a Reddit thread about stronghold distribution, it was reported that Jens (the development lead on Minecraft) said at PAX that strongholds spawn within a 500 block radius, centered around your initial spawn point.
As of 1.8, the number of strongholds spawned per world was increased from one to three, which makes them rare, but not egregiously so like they were in 1.8-pre1.
Combined with the 500 block radius spawn parameter, if you do any amount of exploring, you should hit one by sheer dumb luck.
If you're impatient, there are alternatives depending on the version of Minecraft you're using: Beta
1.9 As of 1.9-pre4, you can use Eyes of Ender to locate the direction of a stronghold by throwing them up in the air.
Once thrown, they'll change direction to point to the stronghold.
Example usage from CaptainSparklez: Beta 1.8
There is no in-game way to find a stronghold, but you can use a cartography program like MCMap
Live (Mac) or Cartograph G (Windows)
(see also this question for other alternatives) to look for tell-tale signs of a stronghold: iron bars and bookshelves.
Here's a render of a world I just generated, highlighting bookshelves (click to enlarge): The bookshelves are just barely visible at this resolution, but they're marked by the green circles.
Enhance!
Bam, stronghold.
The render is from x=-750,z=-750 to x=750,z=750 , so it appears the Strongholds are on the very cusp of the reported 500 block radius limit, if not a little outside.
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There is an in game mechanic for solving this problem.
Use Eye of Ender
If you want to get into the end, be sure and bring a bunch of them though!
gaming.stackexchange.com/users/16398/John
I take the seed ( F3 ) from the world in which I want to find the stronghold, and start a creative map using the same seed.
When you do this, you have an infinite number of eyes of ender to throw (throw by right clicking).
Once you find the stronghold by throwing the eyes of ender, and then following them (you will have to dig to get to it), write down the coordinates of the stronghold ( F3 ), and then go back to your original map.
Head to the coordinates of the stronghold you found, and, if you used the same seed in your creative world as you did your original world, the stronghold will be there.
This method makes it so you don't have to kill lots of endermen to get the eyes ender.
Hope this helps!
gaming.stackexchange.com/users/26785/ghjfgyubtyuyu
It's pseudo-random, though you can probably find strongholds with something like MC Map Live , though it's only for Mac OS X.
Here's the Windows equivalent, though I'm not sure if it'll work for 1.8.
Cartograph .
Just explore your world and manually search for strongholds on the map.
gaming.stackexchange.com/users/11946/Dante
You can always use triangulation if you're inclined to work out to solve similar problems in real life too.
gaming.stackexchange.com/users/8568/Rushyo
Lots of great answers here, but no techniques on practically finding the strong hold with eyes of ender.
The Bottom-Line is, eyes of ender are rare
so you need to use a technique called ' Successive Overrelaxation ' .
In short, when you know the direction from the first throw, just run, and run and run for a long time.
Then, throw again.
And rinse and repeat.
This technique preserves eyes, and it saves some time too.
gaming.stackexchange.com/users/16142/Pureferret
I've been searching for hours in the game to find a Stronghold and haven't seen one. Apparently lots of mobs and treasures can be found there, but the question is where can we find those?  Can we find them near a ravine or abandoned tunnels?
How can I find my World's Strongholds?
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<minecraft-java-edition>
gaming.stackexchange.com/questions/30340
gaming.stackexchange.com/users/11188/Warface
The number of strongholds increased from one to three in 1.8. As of 1.9, you can use Eyes of Ender to locate the direction of a stronghold. You can also use a cartography program like MCMap. A suggested distribution of strongholds as well as certain techniques are useful tools in order to find strongholds.
Certain amount of strongholds spawn in specific areas thus making it easier to find them. Using Eyes of Render along with a certain technique serves as a solution. Several programmes could be used to detect stongholds. Using triangulation could serve as a solution to the problem.