I use the wireless headset for the slim, which is actually surprisingly decent.
Standard headset out of the box should work just fine to be honest. I fear that if you are having issues with that you may have issues with others aswell. Try making sure your NAT is open as that can cause communication issues over Xbox Live. However if it is broken or faulty can you not just get it replaced?
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In regards to using normal headphones you can do the trick of using a component lead with your HDMI lead and using the component lead to run your headphones by using two adaptors. phono to 3.5mm and female/female adaptor so you can plug your headphones in.
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Some specially designed headsets (Turtlebeaches and Steelseries for example) will work from the control pad.