Ah, power armour. A necessary, if hard to come by, part of life in Fallout 4. Although not as hard to come by as you might think. Turns out that those pre-war idiots left loads of power armour just lying around, and you can be the person to find them if you know where to look - and with this guide, you will! We've even included map pictures for the ones that are harder to find.
This being a Bethesda game we're going to assume it's probably not an exhaustive list, and we'll add to it if we find any more. Keep in mind that the level of the power armour that you find will depend on the level you are (i.e. higher level characters will get higher level armour spawning). If you're dead keen on getting the X-01 armour then you need to be level 28 as a bare minimum before you go looking for it.
The Museum of Freedom in Concord
The first place you can get a full suit of power armour is in Concord, which is the first proper town you encounter after you leave the Vault, pausing only as you pass through Sanctuary to speak to Codsworth and look sadly at your son's old crib (miraculously still standing after a couple of hundred years). The speaking to Codsworth part will activate a quest marker that will lead you straight to the museum and trigger the quest When Freedom Calls, which involves the power armour.
The mission is basically the intro tutorial to using power armour and if you can't find it you're ignoring everything the game is trying to lead you to.
From the Brotherhood of Steel
This is another easy way to get hold of a complete suit of power armour, although you do have to hang around with Paladin Danse and join the brociopaths at the Brotherhood of Steel. If you board the airship Prydwen you can officially become a Brotherhood of Steel knight and get given a suit of armour for free. Sorted.
The Big Dig with Bobbie No-Nose
Bobbie No-Nose is a ghoul you can find in Goodneighbour and, to cut a long story short, if you agree to help her you end up tunnelling under the city with a small robot that can destroy loose walls, fighting a bunch of mirelurks and ghouls. Early on in the dig you'll be in a room half full of water with a couple of generators (if you reach the subway tunnels you've gone too far).
There are two sections of wall in that room that you can bring down. The one opposite where you came into the room leads to a dead end inhabited by a King Mirelurk (which is a weird walking-seahorse-looking thing that bears no resemblance to other mirelurks). There's also half a suit of power armour. It's probably not worth it for the effort, to be honest, but it's there.
Atom Cats Garage
The Atom Cats are the equivalent of a gang of 50s greasers who are well into tooling up cars, but since cars don't work any more they're content painting decorative flames on suits of power armour instead. Their garage is north of Quincy and if you jump through their hoops you can get access to a power armour suit and parts. You can alternatively kill them all and take it anyway. Kind of up to you, we guess.
Abandoned shack in the Glowing Sea
North of the Crater of Atom and right on the line denoting the edge of the map (which is, incidentally, a lie, because you can go beyond it) is an abandoned shack concealing a secret government surveillance facility. There's a locked hatch in the floor, but you can find the key in the suitcase of the tragically dead scientist (who is also on the floor).
Head to the bottom of the facility and you can find a partial suit of power armour hidden next to the stairs. If you're a high enough level it'll be X-01, which is the most powerful power armour in the game.
Cave in the Glowing Sea
Stick around in the Glowing Sea and head east from the Crater of Atom, and you can find a suit of raider power armour in a cave. Raider power armour is a pretty crummy iteration of power armour, but it's another full suit to add to your collection (even if, because it's in the Glowing Sea, you'll probably have to use a better suit of power armour to get it).
Wharf near Nahant
Nahant is a small town on the north east side of the map, accessible by a bridge or by swimming out to it if you're feeling rad resistant and can't be arsed to go round the dogleg from the mainland. Just before you get into Nahant there's a wharf (it'll be on your left) with a load of military shipping crates and vehicles left over from the war. If you head round the back you can find an open crate with a full suit of power armour in it. Watch out for any military grade robots that might be hanging around.
35 Court in Boston
35 Court is hard to find, because it's not a location that'll mark on your map, but it's worth your while. It's between two streets in Boston, north of the Shamrock taphouse, and does, to its credit, have a big sign saying '35 Court' by the door. Go inside and work your way up to the roof and be ready to take on the assaultron and sentry bot that guard it. Once they're down you should be able to see a red button in each of the little chambers they were occupying, so press them both to open up a separate blast door. Behind this blast door is, finally, a full suit of power armour, and if you've levelled up enough it'll be a complete suite of X-01, X-01 being the highest tier power armour you can get in Fallout 4.
Pier near the Old North Church
Head to the Old North Church near the river in Boston (you should know it if you've encountered the Railroad) and go north from there. You'll find the riverfront with a number of piers, one of which is covered in a bunch of old military junk and has a shipping container hiding - you've guessed it - some power armour.
Fiddler's Green Trailer Estates
We've mentioned Fiddler's Green before, in the context of it just being a cool place to hang out, but it's also got a partial suit of armour in one of the trailers.
Train car on the road to Boston
After you head out of Concord and start going to Boston you'll pass the Drumlin Diner on your left. Keep going straight down the road and you'll come across a crashed train, just off the to the side. One of the cars has a suit of power armour in it, although it's locked in with an advanced terminal so either be prepared to hack or bring a hacker with you. You should also keep an eye out for ghouls lying in wait to jump at you.
Train car near Tenpines Bluff
A little way north east and down the hill from Tenpines Bluff is another wrecked train car, with another locked terminal and another suit of power armour inside, just like the one on the way to Boston.
Crashed Vertibird near USAF Satellite Station Olivia
A bit north east of the Satellite Station (which is itself at the north of the map), and just off the road, is a crashed vertibird. It can be hard to spot because it's the same greenish brownish greyish colour as the rest of the landscape. Standing just ahead of the nose is some power armour to pick up. No catches.
On top of the freeway near College Square
The section of freeway you want follows the road that sweeps past the big loop in the river, before it reaches Boston. You can head north west from the College Square or follow the road up from the second bridge on that aforementioned loop of river, but either way you'll reach the right sort of area. The power armour is stood on the very tip of the broken freeway, staring out into the wasteland bleakly, a lone witness to the passing of time that will one day rust into nothing itself. So you can go up and nick it pretty easily.
Bottom of the lake near Lexington
Lexington is near the middle of the map, and you're looking for the medium sized, sort-of-oval lake to the east of it. At the northern end of that lake (stay with us) you can just see the tail of a crashed vertibird poking out, if you dive in that spot you can find a sunken suit of power armour. Just be prepared to hold your breath until you get into it.
Military sheds near the Greater Massachusetts Blood clinic
The Blood Clinic is on the west edge of the map, basically slap bang in the middle. Between the Clinic and the lake below it is a group of unmarked and disused military sheds, one of which had a suit of power armour in it. There's also a power armour station. Nice and easy.