Fringe on a Non-Transparent BG

depends.  but one way would be to ctrl click the layer to load it as a selection, add layer mask, then use the Mask properties panel. contracting the mask, would hide the fringe, for instance.

you could also use a hue saturation layer if the fringe is coloured. it's too painful typing on this iPad to go into details right now.

Isn't a layer over a transparent background exactly the same as a layer over an opaque background, with regard to the layer itself?

My first thought is to use a selection, then expand/contract it at the edges and introduce more transparency (either by masking or erasing pixels) to hide the edges.  You could even use Refine Edge.  But you haven't given enough specifics to make detailed advice possible, really.

Perhaps you could post an image or two showing just what it is you are trying to solve (don't skimp; show the whole Photoshop desktop).  That might help net you more specific answers.

-Noel