There's nothing more frustrating when working with boiled eggs than fighting with peeling them! I learned this trick about 25 years ago and its still my tried and true'd method for boiling eggs so the shells practically fall off on their own!
Start by leaving your eggs in the refridgerator and getting your water to a rolling boil. Once your water is boiling, lower your COLD eggs into the BOILING water. A pasta spoon works awesome for this!
You'll want your water to stay boiling, so I put a lid on it and lower the temp down to about 1/4 (3 on a scale of 10) after it comes back to a boil. Set your timer for 20 minutes.
At this point in my picture taking I turned around and discovered I had a helper so I'm gonna include her picture here for laughs :)
About 2 minutes before my timer is going to go off, I put about 2 cups of ice cubes in a bowl and add some water. The trick here is to take your HOT eggs and instantly cool them off with ice water. I've tried water out of the tap and it just doesn't work as well.
I let my eggs sit in the ice water for about 2 minutes (about the time it takes to rinse my pan out, dry and put away). I then take an egg out, tap it on the side of the sink and the shells basically fall into 2 pieces. Rinse the egg just in case any shell pieces are left behind and move onto the next egg. I peeled these 6 eggs in about a minute and a half (and that was with taking pictures).
In this batch, I didn't "lose" any eggs, but its normal to have 1 or 2 crack in the boiling water... depending on what I'm making I usually just chop those up fine and put with the egg yolks (for deviled eggs or salads). Because of this, I usually add 1 or 2 eggs more than I want to end up with. Clean up is simply roll the paper towel up and toss.