My 06 S-Type w/155k mi. just started whistling at me when in reverse.

 

...and I assume not only, when you are in reverse, while driving next to a train station, while the train gets the signal that it may depart from the station?

I would not think it makes a difference for the engine to drive forward or backwards, which would reduce the possible source of the "train-conductor" in your S-Type to the transmission or the brakes!
Hence, you might want to check your ATF level (there is a procedure for it, WHILE the engine is running with the drain plug briefly opened (then then closed) while running...
And it never does any harm to check for the condition of the brakes, i.e. removing the wheels one after another. Brake pads still thick enough? Has a bit of copper pasted been applied when renewing the brake pads last time? Are the rotors still in their legal thickness limit?