For over a year now, I've been eating three boiled eggs every morning. It's not that I particularly like my eggs boiled: I prefer them scrambled, and I make good scrambled eggs. But boiling eggs is easy, and one of the secrets of my dietary success has been the rule, "if it ain't easy, it ain't happenin.'" I start them boiling, and by the time I'm done with my oatmeal and brocs, they're ready: the part of my breakfast I most enjoy.
So I have a lot of experience now with boiling eggs, and I feel I can make authoritative statements about it. In particular, how to do it so as to make peeling them easy.
First, and most important: don't boil new-laid eggs. Just don't. There is no way to make really fresh eggs easy to peel. (And anyway, if they're as fresh as that, why wreck them by boiling them? Make an omelet.) I deliberately age the eggs I'm going to boil: I want them at least a week old.
Second: "shock" them with cold water. This helps, a little, though not nearly as much as some people think. Mostly it just makes them easier to handle when they've just come out of boiling water.
Third: some eggs will never peel easily. Ever. No matter how old they are, or what you do to them. The occasional egg comes along with a membrane that sticks tighter to the flesh of the egg than to the shell, and there is no good way to peel it. Surrender gracefully. You can strip the first layer of flesh off and scrape it from the shell with your thumbnail. Whatever. Do what you must. It's not your fault.
It's delightful when the shell slips off an egg all of a piece, and it makes you feel very skillful and clever. The impulse to take credit for it is overwhelming. But in fact it's just dumb luck. Anybody would have found that egg easy to peel. If you only boil eggs once in a while, you can be forgiven for thinking that you've got the hang of it now and you've solved that egg-peeling thing and you know exactly what to do with them. It's an illusion, I'm afraid. There is an impossible-to-peel egg in your future. Maybe not today, or tomorrow, but soon.