A Complaint is a Gift
When it reaches a level of authority within an organisation it can take a slightly different tone.
I fully subscribe to the "Complaint is a Gift" methodology of complaint management. If you haven't heard of it before, then google it, and get the book. It's an excellent guide to how to turn a complaining customer around.
Firstly. The key thing to remember is, by them contacting you in the first place, they are holding enough trust in your business to resolve the matter for them. And the issue of their complaint is an opportunity to potentially improve your business. So, one of the simplest things that a supervisor can do to immediately calm a customer down is to thank them for taking the time of expressing the problem to them. Something like, "Firstly, mr jones, Thank you for sharing this issue with us. We don't always get it right and its important for our customers to have faith in our ability to resolve issues like this". In any other situation, when you say "thank you", the other person will say "Your welcome". This is a good first step to working cooperatively to resolving the issue.
The next thing to explore is to determine what the customer is looking for regarding resolution. it could be a refund, a replacement of an item, it could simply be the fact that they want to talk to someone, and they may not necessarily be looking for compensation of any kind. But once the matter has been clearly defined, it is usually worth while to ask the caller how they would like this resolved. This is not as volatile a question as you may think. Most of the time the caller will simply want the matter resolved quickly. Sometimes it can get trickier than that... it really does depend on the circumstances and is beyond our ability to flesh it out in this forum.
This is just a start to try to help diffuse a customer. However, studies have clearly shown time and time again that if you can successfully resolve a complaint with a customer, you will likely create a longer term advocate of your business. This will have a very positive effect on your customer retention and customer delight.
Please let me know if my ideas have helped you.