For technical reasons, 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.
This app uses some relatively new web technologies. As a result, some features may be disabled for older browsers. I recommend using the latest version of a major browser such as Chrome, Firefox, Edge or Safari.
The first thing you should do is select a seed and version. 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. The app does this the same way Minecraft does, so it's safe to use letters (and other characters) as well.
Once you entered your seed and version 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 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.
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.
Thanks to Earthcomputer for his work on bedrockified, which made it possible to support Bedrock Edition.