Here's a couple of quick examples of the second useage that Paul mentions, meaning that something did not do very well, or even failed. “The gymnasts did their best but their efforts to impress the judges fell flat” “His jokes fell flat and nobody laughed”.

I agree with Neal, great examples! Particularly "His jokes fell flat..." is a common context. This may just be southern (crude) US English, but if someone literally falls flat, it's more common to say "(s)he fell flat on his/her ass" than "(s)he fell flat on his/her face." Crude but common!

Yes you can use as a description like you fell flat on your face but its more commonly used as an expression to fail

...on second thought, it really depends on what direction you fall, forwards or backwards