In Minecraft, every abandoned mineshaft has a single starting point. It's usually a room with a dirt floor (may intersect with other map features like ravines) and several exits to corridors. The starting points are predetermined by your map seed. In average, every 100th chunk has a starting point for a mineshaft. These chunks are highlighted in brown. The colors of the surrounding chunks are a very rough approximation of the chance to find a piece of an abandoned mineshaft in each chunk. Thus, the most reliable way to use this app is to walk to the center of a chunk with a starting point and then search for the room on different layers. The height level of the room is less than y=53 (random).
Around the origin point of the map, the chances for abandoned mineshafts are actually smaller. The chunk with the coordinates (0,0) has a 0% chance to contain a starting point. The chance increases linearly in every direction until it is at 1% as soon as either chunk coordinate reaches an absolute value of 80 (1280 in standard coordinates).
How accurate is it?
As mentioned before, the chunks highlighted in the brighter color are merely assumed to contain a piece of a mineshaft. The starting points are very accurate though. However, it seems that under some rare circumstances the generation of a mineshaft can fail.
Does it work in superflat worlds?
Yes, as long as mineshaft generation is turned on and the spawn chance is unchanged.
For the reason mentioned above, you need to know the seed of your world to use Mineshaft Finder, unless, of course, you want to find a seed for a new world. If you're playing SSP, the app is able to fetch the seed from your savegame. Alternatively, you can use the /seed command ingame. In SMP, you can use the same command if you have sufficient rights. Otherwise, however, you're dependent on the server owner, who started the world and has access to the savegame and config files.
Mineshaft Finder uses some relatively new web technologies. As a result, some features may be disabled for older browsers. So I recommend using the latest version of either Firefox or Google Chrome to avoid any limitations and to get the best performance.
The first thing you should do is select a seed. You can either type it in manually, or you can load it from your savegame. The latter can be done by clicking on "Load from Save..." and selecting your level.dat, or by drag&dropping the level.dat file into your browser window. Level.dat is a small file located in the folder of every Minecraft savegame. You can find the savegames in the saves folder of your Minecraft installation. On Windows you can use %appdata%\.minecraft\saves to get to that folder.
Levels allow you to store and reuse seeds on this website, without having to load your savegame every time. When loading a seed from a level, the seed will automatically be stored as a level.
You should also know that a seed is always a number (up to around 20 digits). If you type in anything else (like letters), it will be converted to a number. Mineshaft Finder does this the same way Minecraft does, so it's safe to use letters (and other characters) as well.
Once you applied your seed using the "Find Mineshafts!" button, you can start using the map. To scroll, use your arrow keys while your mouse cursor points at the map, or move your mouse while holding down down the left mouse button. You can use the slider below the map or your mousewheel for zooming. The lower inputs allow you to go to a specific point (e.g., your base) of the map and set a marker there. You can always remove and add the marker by double clicking on the map. The "Save Map" button allows you to save the currently shown map as png image file.
Below the map you're also able to switch between PC and Console Edition, which will change the way the highlighted chunks are determined, but only if there's actually a difference between the two platforms. Also, with Console Edition enabled, the map will be limited to its actual size.
When using a touch-enabled device, an extra option for enabling/disabling touchscreen control will appear below the map. With that option enabled, you can drag the map with your finger to navigate, you can pinch to zoom in and out, and you can tap and hold to set a marker on the map. By double tapping on the map, you can quickly enable/disable the functionality as well.