Cool thread.
Transformers have a lot of "things" inside that may or may not reduce hum and noise. Try some experiments with the leads and can grounds. never know which will be best.
"things" from the center and moving outwards:
1) core ground-usually a wire solderd directly to a lam, which is then brought out to a terminal connection that may have other grounded "things" attached to it. Or it may be soldered internally to the can.
2) shield between pri and sec- can either be foil that either has a lead soldered to it or just a wire pressing against it held by tape, or the shield can be a bunch of turns of copper wire that are un-terminated on one end, this is your typical UTC shield. The wire is brought out and tied to the core lam ground in most cases, so there is really no way to use the core ground and e-shield ground seperately in this case
Note that foil shields must not have the ends touching, as this could turn into a big shorted turn that the primary would see, which might make the transfomer useless.
3) outer foil shield like API-looks cool but virtually useless.
4) internal shielding cans- these can be real mu metal, or imitation stuff, like they use in UTC.
Triad.... UTC....Telefunken etc sometimes use copper and/or brass foil or thicker copper sheet metal inbetween mu cans for extra boogie woogie.
5) core brackets-not used much in line level stuff but sometimes seen on after market outputs, like Sowter. Used a lot on power outputs. Not the most reliable ground because of paint, metal oxidation, lam coatings, loose bolts, etc.
6) external case- nice steel cans like UTC LS series, round mu cans like most stuff being made today, extruded rounded rectangular steel like old Triad and Peerless...usually a wire is soldered to the can directly or soldered to a lug that had been spot welded to the can. So you have two possible paths to ground here, thru the case mounting hardware, or "terminal 11"...
the "earthy" stuff is most important at mic input levels, important at line in and innerstage levels, and not as important in output levels.
Sometimes the only way to have access to all of these earthing points is to wind your own transformer. Some x-former makers have seperate grounds for e-shields amd cans.