[Last updated: 1/9/19]

The way to unlock certain robots has been asked, especially MM3 robots, has been asked a surprisingly high amount of times.

So, this guide is a full guide on how to unlock every robot, MM1+ robot masters, and things like Roll, Disco, and Rhythm.

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1. The Mega Man 1, Mega Man 2, and Mega Man 4 Robot Masters.

I'm grouping these all together because they are all unlocked the same. For the MM1 Robot Masters, use only neutral type attacks (Buster Shot , Mega Buster, Mega Ball, and so on.) on the Robot Master of the stage. Metal Man , Hard Man , Fire Man , Bright Man , and so on are all unlocked with this method.


NOTE: It is okay to use elemental attacks on the mechas of the stage, but you cannot use elemental attacks on the actual robot master!
NOTE 2:: The stage has to be the Chapter 2 Robot Master stages. The fusion stages don't work for unlocking them!
This method is the same for Chapter 2 of Dr. Wily, and Dr. Cossack.

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2. Roll, Disco, and Rhythm

These three are unlocked the same way essentially, but with a slight variation.
For Roll, using Doctor Light, lose any battle. For new players I recommend going to Elec Man's stage with Mega Man only.
For Disco, it is the same method, but with Doctor Wily, and you have to lose two different battles.
For Rhythm, it is the same method, but with Doctor Cossack, and you have to lose three different battles.

NOTE: The failed missions have to be different missions.



UPDATE:
This old method still works, but there is now a new method:
To unlock Roll, Disco, or Rhythm now all you have to do is beat the Intro Field a second time as Light, Wily, or Cossack respectively
So now you can unlock all robots without having to ever lose once.

There is also an extra method for those that don't want to do either of these methods, and that is the passwords. To unlock the password screen, click the doctor sprite about 10 times in the tab above the home/shop/robots/etc tab and to the right of the battle points tab. Then you get a secret password popup. From there;
To unlock Roll from this; put in "Robot Get: Let's Rock n' Roll!"
To unlock Disco from this; put in "Robot Get: Panic at the Disco!"
To unlock Rhythm from this; put in "Robot Get: Rhythm and Blues!"

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3. Mega Man, Bass, and Proto Man


These three are unlocked automatically with you unlocking each doctor. Start the game and you get Mega Man. Unlock Doctor Wily and you get Bass, and unlock Doctor Cossack and you get Proto Man.

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4. Mega Man 3 bots, and MM5+

All you have to do is just scan the robot you want (Snake Man, Magic Man, Jewel Man, etc) and you should be able to buy them for 48k zenny in Kalkina's shop. That's it. No having to buster only them in their fields or anything. This method is incredibly easier, and I personally recommend it, but if you don't have much zenny, either sell some items or scan your stars. Besides the large amount of zenny you need, there is not much challenge or confusion to this method at all.

List of robots you can unlock through buying them (updated 1/9/18):
-All Mega Man 3 Robot Masters
-Wave Man
-Star Man
-Gravity Man
-Gyro Man
-Splash Woman
-Magic Man
-Slash Man
-Jewel Man
-Blizzard Man
-Freeze Man
-Frost Man
-Block Man

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Hope this helps!

FINAL NOTE: Any robot that isn't mentioned here is NOT unlockable as of right now. When new robots become unlockable I will try to quickly get out an update for how to unlock them as quickly as possible. As MegaBossMan noted below, some users have figured out how to manipulate their user data to get robots that aren't unlockable yet. I will not add a guide on how to do that yourself, as I don't know how to do that, and I don't want more people breaking the game.[/i]

edit: thanks for the sticky, mod squad *finger guns*
edit 2: an explanation of how to unlock the new bots in the most recent update as now been added
edit 3: various fixes and cleaning up to this thread; field method for unlock mm3 bots has been removed