My iPhone 4 doesn't make a peep, the volume controls, mute button or headphone jack have stopped working!

A phone that doesn't make a sound is not much of a helpful companion. Luckily there's a connector for everything to do with sound (except for the speaker on the bottom of the device). It's the one pictured below. Make sure it's connected to the logic board.

If the speaker on the bottom of the phone seems to be broken, check if this large connector is plugged into the board.