I know what you are thinking, "Why would I want an alarm to tell me when the power is out?" Well, I want to make one for the freezer. The hot tub is on the same circuit and I've had the circuit breaker trip and I didn't notice until the next day. Luckily I noticed in time.
I could design a circuit but it is a lot easier to use something that already exist if it is cheap enough. I plan on getting one of those plug-in rechargeable flashlights that light up when the power goes off. Then get a piezo buzzers from Radio Shack (the Source) and hook it in parallel to the light bulb. That way, the light comes on and the buzzer annoys us until we flip the breaker.
This would also work if the freezer is being run from battery and inverter. What happens when the batteries get low and inverter automatically shuts off? This would warn me to do something else. Nothing worse than losing a freezer full of meat.
And, of course, some people would use this as an alarm to let them know that power just went off while they are sleeping.