The best way to fight each phase is dependent upon the category you're running and what equipment you have.
This long video goes over all parts of the Majora fight in detail, discussing strategies and how to react to various scenarios. It is geared towards categories that don't have Fierce Deity Mask or Great Fairy Sword, such as Any% NMG, Any% Glitchless, and All Dungeons Restricted. For other strategies including those involving Fierce Deity Mask and Great Fairy Sword, look below on this page, but general information in this video is still useful in any category.
Mask hovers around the room and has two attacks:
Spinning attack: This attack is variable length and can last quite a while. During it, Mask can be still be stunned, most easily by either a spin attack, an explosion or Deku coming out of one of the flowers on the floor of the arena. This attack may either deal 1 heart of damage or simply knock Link down without damaging. The outcome depends if Link touches only one or both hitboxes that appear around the Mask.
Laser: This attack has fixed length. It can be reflected with the Mirror Shield, but it deals fire damage, so Link will void out if hit with it as Deku or Zora. If hit with the attack as human, make sure to not stand still, in order to avoid further damage ticks from the fire and to extinguish the residual flames before transforming to Deku or Zora, as otherwise you'll void immediately after the transformation.
Majora's beam attack, if close enough to Link, will instantly lock onto him if you press the Shield button (including as other forms!), so never press and quickly release Shield. Other things to note are that Goron gets stunlocked by the beam until the attack stops and Majora never initiates that attack on Fierce Deity.
If you really want to get rid of the Remains, which have 6 health each, and you have Mirror Shield, reflecting Majora's laser against one of them for 4 frames will instantly kill it.
Depending on what strat you want to use, you can initiate as either human or Zora. You may also want to get the Magic and/or Arrow drops in the jars behind the starting position.
Below is an approximate transcription of Morpheus' video, regarding starting as Zora.
Zora can be used if you don't want to use the human strat with the Bow for the Mask phase, which despite being faster, is also more precise. That will be explained on the section below this one.
As Zora, once you gain control, hold forward, hold B to get the Zora fins out, target the boss and throw your fins. If they successfully stun, switch over to human, shoot a Light Arrow and pull a bomb while walking towards Majora. Throw the bomb when it starts quickly flashing, that should get Majora stunned again. Then shoot another Light Arrow and turn Zora once more to start the second part of this phase.
Same as before, hold B to get the Zora fins out, target the boss and throw them. Switch to human, shoot Light Arrow, pull bomb, stun the Mask with the explosion and shoot the last Light Arrow of this Mask phase.
Notes:
If you notice your fins aren't going to hit, there is something you can do to try again quickly. If you land from a jump slash while holding B, Link will be able to throw an extra pair of fins even if your first pair is already in the air. Since you'll be targeting Majora the whole time, you can keep jump slashing and throwing as many fins as you'd like. HOWEVER, if you actually manage to stun the Mask with your first pair, the second pair you threw will do 2 points of damage upon successful contact, which is less than the Light Arrow's 4 points of damage.
Another thing that can happen is your Light Arrow might go straight through the Mask. You can try taking another shot in case the first one was quick enough, but you might have to wait for the Mask to start flying again and re-stun it.
See second half of video for steps and visual cues.
Below is an approximate transcription of Morpheus' video linked on top of the page, regarding starting as human.
If you want to save more time compared to using Zora, it's faster to start as human and walk into the fight without Bow in hand. It must be pulled out immediately after the fight starts.
You need to mash Light Arrow, hold target and aim up until this first frame when the camera fully pans out behind Link, at this point you can let go of everything and, if done correctly, that will stun the boss. Then you should roll up to the Mask, draw Link's sword and jump slash it, then pull a bomb and throw it when it starts quickly flashing, which should stun Majora again. Lastly, shoot another Light Arrow.
To continue this strat, you should move Link up to where the Mask is before the second part begins, use Light Arrow in first person, aim up, target it and shoot its back part, which will stun it if done correctly. From here you can damage it with Light Arrow, stun it with a bomb and damage the Mask with Light Arrow one last time.
Notes:
Stun with a sword beam, then jumpslash and back away before the cutscene. Stun with another sword beam and hit with another sword beam. Before the transition to Incarnation begins, you can turn around and try to hit one or more of the remains with sword beams.
A bomb exploding in the center of the room will always knock Incarnation down. Otherwise, its actions are erratic and include moonwalking, a light ball attack, pretending to trip, and sprinting. Damage can only be dealt after knocking Incarnation down; the attack that knocks it down doesn't deal any damage. Once it falls, it's stunned while writhing. The next hit deals damage and makes it get up and hop, at which point it has a short period of additional vulnerability before it runs away.
It's possible to immediately knock Incarnation down again after this period ends, but only with the right timing and with certain attacks, as most damage sources make Incarnation's hurtboxes disappear until after it runs away. Relevant attacks that allow for stunlocking because they don't make the hurtboxes disappear include Kokiri sword spin attacks, Deku spins, and deku stick damage (including when done with power crouch stab using Kokiri sword). Every Deku sword attack removes the hurtboxes.
Stun with a sword beam, then spam sword beams.
Wrath can dodge or block most attacks, but it never dodges a light arrow shot, even if you're out of magic and Link only fires a normal arrow.
The simplest tactic is to stun Majora with a sword beam, then land 5 jumpslashes with animation cancels. The strategy in the video below is faster and more complex, and is explained in detail in the video.
This video demonstrates the fast strategy with input display.
This strategy loses about 3.6s to the optimal strategy with Fierce Deity. Take a step forward and then roll. Target Wrath and mash B during the roll to slash it. Transform to human, take out the GFS, and jumpslash Wrath. Power crouch stab four times to finish the fight. In order for the crouch stab stunlock to work, each crouch stab except the final one must be timed just before Wrath would otherwise get up and jump away. A good visual cue for timing these is inputting another crouch stab each time Wrath's hands reach the floor.