What causes old power supplies to start humming?
A transformer, as you well know, is made up of two or more coils around a core of ferrous material.  That ferrous material is not a solid lump of metal, but a series of plates laminated together with adhesive.  This is done because:


  Early transformer developers soon realized that cores constructed from solid iron resulted in prohibitive eddy current losses, and their designs mitigated this effect with cores consisting of bundles of insulated iron wires. Later designs constructed the core by stacking layers of thin steel laminations, a principle that has remained in use.
  
  -- Wikipedia


So you have lots of steel plates stuck together, but not only that:


  Each lamination is insulated from its neighbors by a thin non-conducting layer of insulation.


Lots of metal plates, each with an induced magnetic field. That magnetic field acts between the adjacent plates stretching and squeezing the adhesive and insulation between them.  Over time that adhesive starts to break apart and the laminated layers separate from each other slightly.  This is the humming noise you can hear.  It's always present, but once the adhesive starts to break it gets louder.  These micro-fractures in the adhesive may not be visible to the naked eye, but in extreme situations they may be so bad the layers of lamination become loose and the transformer literally rattles as you shake it.

Also, the more current you draw through a transformer the larger the induced magnetic fields, and thus the louder the transformer hums (and the shorter its life span).