Finding a saved file on the Simulator?

Yes the you can access the Simulator files from ~/Library/Application Support/iPhone Simulator/.

In this folder are the various iOS version numbers so pick the one you are running. In that will be Applications and you'll see a bunch of GUIDs, one of which will be the running app.

One thing in OSX 10.7 is that the Library folder is now hidden. This article explains how to restore the visibility; Show User Library Directory in Mac OS X 10.7 Lion

The location changed for Xcode 6 with iOS 8:

~/Library/Developer/CoreSimulator/Devices/DEVICE_UDID/data/Containers/Data/Application/APPLICATION_ID/Documents

DEVICE_UDID: can be found out in the Device Inspector (Window -> Devices) and Select the simulated device you're running on. The field Identifier contains the device's UDID.
You can also get a list of the simulator UDID's by running the command xcrun simctl list (thanks @JeremyHuddlestonSequoia)

APPLICATION_ID: I didn't find this number somewhere, so I looked into the folders until I found what I wanted. If someone finds out where this number comes from, let me know.
EDIT: The ID of the application changes every time you start the simulator, but when you keep a finder window open pointing to that folder, you'll always find yourself in the correct folder (no idea how they do that; every other application seems to loose track of the files), which is extremely handy!