My toilet makes a kind of whistle sound through out the day. It is not a specific time or not necessarily happening when I flush. It can happen at any time. Even when we have not used it for hours. :confused:

There are a few things that would cause this problem. First make sure the lever is free and not hanging up letting water seep past the flapper. Then check the linkage between the flapper and the flush lever, When the tank is full you should have at least 1/4" of play. I have found this vibration occurs when a very small amount of water is seeping past the flapper causing the ballcock to open just far enough to cause vibration between tha ballcock washer and the seat causing the pipes to pick it up and amplify it. This noise will stop when you jiggle the handle thus seating the flapper firmly. You might also check the edge of your flapper. If it has wavey edges, replace it. Also this can happen if the tank level's too high and the water's leaking into the overflow tube. Check the tank level. it should be 1/2" below the top of the overflow tube.
Good luck, Tom