Does anyone have any tips for reducing or eliminating the loud humming noise from an American Flyer transformer?
The one in particular that I have noise trouble with is a #22035 transformer. I have tried tightening the two screws down that holds the coil to the base, but with little success in noise reduction.
Sorry, don't know anything about that particular transformer, but one of the go-to DIY techniques I've seen for reducing mechanical noise is vibration isolation. Tightening the mounting screws *will* eliminate any rattling from a loose junction, but is also more likely to better transfer vibration to the larger "sounding board" of the case.
Would it be possible to create (or enhance) rubber mounting dampers of some sort? I have in mind rubber rings between the coil and base, and between the head of the screws (or washers under the heads) and the base on the outside. You may need to enlarge the holes in the base to avoid the screws' contact with the base, and you want to make sure the outside heads of the screws do not rest on the layout. Ideally (if you can find some), you could use those little one-piece mounting 'donuts' that are designed to pop into holes in the case and provide isolation inside, outside, and between the screw shafts and base.
Also, there should be additional acoustic isolation between the transformer and the layout supporting it, usually by way of cone-shaped rubber 'feet' with mounting screws in the middle, but in a pinch by just resting the transformer on a half-inch or so sheet of foam (be sure not to obstruct any cooling vents).
About the only other measure I can think of would be to add dampening padding inside the case, especially on any sheet metal surfaces that might otherwise add sounding board surfaces that can radiate any internal vibration, if there's enough clearance. Car manufacturers make good use of self-stick sheets of such sound deadening material on sheet metal surfaces, and if you can lay your hands on some (from a body shop, etc.) you may be able to damp out some additional vibration/sound. In any event, good luck!