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Power armor has to be found on the world map in order to upgrade your suit. While it may appear otherwise, there are very few "guaranteed" spawns - most of the time, you will find a power armor frame with randomly generated, level-dependent pieces present. Which pieces spawn is determined by your character level and the level of the cell. In general, T-60 and X-01 power armor spawns are limited to cells which have a high minimum difficulty level set (such as the S. Boston military checkpoint, which has a 25-30 cell level) and the level, according to the following table:
Note that entering a cell that contains a spawning power armor will lock that cell to that level, making it impossible to get a higher level armor. Which also means you might want to restart from a certain save file before going there again. If you wanted X-01 armor you would have to be around Level 28.
In the back near "finish line." It is marked as owned, so the player has to steal it. The optimal way is to either sneak or attract monsters to the front and get it when they are distracted.
The player will not be able to fast travel until the combat ends. It's impossible to verify if the player is successful until the Atom Cats deal with the threat.
Alternatively, the armor can by purchased from Rowdy (expensive). Killing the Atom Cats will preclude purchasing their custom T-60 paint job.
A raider spawns at the bottom of the quarry wearing T-51 power armor (potentially different levels available here).
T-51 power armor.
Scrapyard (Hub City Auto Wreckers)
Gunner commander.
T-45 power armor minus helmet.
Quincy
On the Gunner commander.
T-45 power armor minus helmet.
Outside Switchboard (first Railroad quest, roughly between Super Duper Mart and Lexington)
There is a catwalk between two buildings just outside the Switchboard location behind a big Slocum's Joe billboard, the raider spawns up on that catwalk and has a missile launcher (potentially only at higher levels).