Past two days: Six eggs from 8, 2.5yr old, molting, hens. All six peeled cleanly.  Have used both distilled water, as well as the hard stuff from the tap.  Have never bothered with salt.
Crack the eggs (delicately - break the shell and not the membrane) the more fractures the better.  Place eggs in cold water.  Heat eggs slowly for ten minutes or so (cooks layer closest to shell).  Then crank to boiling.  A few eggs will leak a tiny bit anyway, but most will come away cleanly from the shell (still hot - or rinsed in cold water).
Another method is to just boil the suckers.  Rinse them until they're cool enough to handle and then crush the shell (start with both hands, then just keep rolling and squeezing in one hand).  Don't squash the egg!  When the shell takes on the appearance of a junkyard windshield, peel it away.
During our `pullet/jenny' summer we had more eggs than we knew what to do with - I wasted more eggs (trying to peel them) than were thrown at Spiro Agnew.  Then I boiled a turkey egg that I'd dropped on the floor (tough customers with a tougher membrane) and cracked.  It was the only one to peel cleanly after boiling.  Have been honing the method ever since.
90% success rate (conservative estimate)