SUPER SAIYAN 1/2/3
Let's start with the beginning: the original Super Saiyan. The form was first achieved by Goku after Freeza killed Krillin on Namek. It's look is as iconic as it can get, being almost synonmus with Dragon Ball as a whole, even though it only appeared at about the half-way point of the story. It's requirements have been specified to be: a pure heart, a certain unspecified level of power and a huge amount of rage.
I won't hesitate to call the design brilliant, despite it's simplicity. It conveys everything the author meant. Goku has become something else which should be the first point of a TRANSFORMATION. The hair stands up, embelished in in a golden aura, turning it blonde. The pupils change their normal color to green and the body becomes all jacked up and all muscles tense.
However, the main differences (raised hair, tense muscles, different aura) are not the first time Toriyama chose to use those specific characteristics to imply an increase in power. During the Namek Arc, Goku had trained his body enough so that he could increase the duration of his, then, only power-up/multiplying technique: the Kaioken. While in the Saiyan Arc, during his battle with Vegeta, Goku had only gotten around to a x4 Kaioken, by the time Freeza appeared as the main threat, he had already mastered a x20 multiplier of his power. This was specifically chosen to be represented by the author when Goku first encounters Captain Ginyu, for whom he displays the upper limits of Kaioken that he broke. And, while at the lower levels the Kaioken only appeared as a red aura surrounding the body, with the muscles tensing, at the upper levels we are presented with the same Super Saiyan characteristics: the upwards hair, muscles so tense they look as if they are about to burst and a huge aura covering the body.
All in all, the Super Saiyan transformation is a great design, borrowing from what previously came before, while twisting it just enough to make it it's own special thing.
Let's now move on to a design which has ascended past the design of a regular Super Saiyan, or the Super Saiyan 2 design. This form was originally achieved by Kid Gohan during his battle with Cell, after the villain killed Android 16. At the time it was supposed to showcase Gohan's incredible fighting potential coming out. And it showed:
Though the form was unlocked in a fit of rage by Gohan, it was never stated that anger is needed any longer to do it, (rage boost being Gohan's way of unlocking his inner strenght since he was introduced implies only a high enough level of power is requiered). This meant pretty much that any Super Saiyan could become a Super Saiyan 2 with enough training, so, of course, Goku and Vegeta achieved it too. Sadly, it's first appearence is the only time it is really distinguishable from the regular Super Saiyan. That's partially because characters like Vegeta (and Gohan with his new haircut) always had the hair standing up, meaning the only truly substantial thing that makes it look like a different transformation from Super Saiyan is the lightning, as the spikier hair is barely noticeable. That was also problematic when the anime decided not to always have the lightning in the aura, therefore giving birth to a lot of debates whether a character is in one form or another. The only one managing to escape that was Goku, as Toriyama was smart enough to differentiate the number of bangs Goku keeps in each form (4 in SSJ, 1 in SSJ2) though even with him there has been confusion.
However, it wasn't long until even this level of power was surpassed and Toriyama had to come up with the Super Saiyan 3. The form kept the lightning of the previous transformation and the recognizable golden hair-green eyes combo, however, there is no confusing this one with any other. The transformation was first achieved by Goku in the Other World, specifications for it not being given (though Gotenks achieves it all the same after a couple of hours in the Room of Spirit and Time, so we can assume a high enough power level is the only requirement). It's main characteristic is, of course, the absurdly long spiky hair. However, one small touch up always gets to me. The lack of eyebrows.
What purpose could lack of eyebrows have? Well, with a master stroke, Toriyama returned the Saiyan to his original transformation introduced ever since the very first story arc. The Great Ape.
Even so, the Super Saiyan 3 design was a simplistic design that managed to convey the even further increase in power, with only a few touch ups, however, without falling into the same trap Super Saiyan 2 fell, where it was nearly indistinguishable from the previous form.
I know I skipped the grade forms, but those were just bulkier Super Sayians anyway and made little impact, not the mention that whatever could be said about them is already commented upon in-universe during the Trunks vs Perfect Cell fight. Let's move on.
SUPER SAIYAN BLUE
As a design, there's not much that can be said. If you like Super Saiyan and you like the color blue, this might just be right up your alley. I will admit I am not the biggest fan of the form, but I am willing to embrace it, though I do think the previous transformation (Super Saiyan God, which we will talk about later) was a much better looking transformation all around.
The form is now powered by God ki, unlike regular peasant ki. This, as mentioned in the new series Dragon Ball Super, makes it a much calmer and easier to use/maintain than the regular Super Saiyan form. In that sense, I do think that the color choice to represent that is the best one possible. The blue is not dark, implying some other more intense state, but not the lightest nuance either, implying it's still battle ready. In that way, the form manages to convey trough just it's main change, the main reason why it's supposed to be a better transformation in-universe. So, in that sense, the form gains some extra-points.
Other than that, there is not much that can be said about the form. However I would like to explain why I said at the beginning why I am willing to embrace it and how it stands in relationship to this:
This caused many fans of the series to hate such transformations and imediately dismiss them as nothing but uncreative pieces of trash. However, after many years of dismissing pallete swaps as uncreative, when the main series gets reinvigorated and it turns out to do the exact same thing, a lot of fans jumped right on the paint-job train. That looks hypocritical to some, or even worse, downright stupid, which never leads to anything good.
But why would fans jump on the Super Saiyan Blue train, yet flip off the Super Saiyan 12 train? I can't say that I speak for everybody, but why I am in the aforementioned train for me can be summarized in one word: context.
In the large majority of cases, when you would stumble upon a ridiculously numbered Super Saiyan form, no matter the author, or the artist, the context of those recolors would always be: user must obtain a certain level of power or user must be very very angry or a combination of both, sometimes worded differently. This point might also help you understand why I specified how every Super Saiyan form designed by Toriyama was achieved. The logic of high levels of power or anger is an internal logic of the show, however, the progression in the fan-designs of the characters does not follow that logic. Toriyama always made minimal changes at best, ever since the Kaioken (in color, hair or body mass) and how one form evolved made sense with the previous ones.
By contrast, when Dragon Ball GT came along with it's own Super Saiyan design, it decided to move on from the logic Toriyama presented and create it's own. As such, Super Saiyan 4 had a lot of requirements that were neither breaking a limit of strength (in the sense that by breaking the limit will give you the form, cause you do need to have Super Saiyan) nor an anger issue.
The internal logic of the transformation doesn't fully work...
But let's not lose track of ourselves. The main point of this detour was to illustrate that introducing new colors or callback to old elements is not in itself bad or to be condemned, as long as it doesn't contradict what came before it.
In that sense, Super Saiyan Blue does not break the logic. As already mentioned, the form is powered by God ki (what matters is it's a new type of ki). In it's first appearence is literally explained as a Super Saiyan with the power of a Super Saiyan God. So, actually, this new transformation just happens to be a freshened up old transformation. It still is the Super Saiyan we knew with the standing up hair and blue-green eyes, however, while the regular ki caused the hair to be embelished in a golden aura, the God ki embelishes it in a blue aura. The form does not break the internal logic of the series, instead creating a new logic for itself, the same way Super Saiyan 4 did.
By this, I conclude that the Super Saiyan Blue form is a notch above the "fanfic" transformations. This doesn't make it an amazing design by any means or a very creative one either, so you won't hear me calling it the best the show has to offer, but it's not the garbage some make it out to be. It's like that college/school student you hate: lazy, but smart.
SUPER SAIYAN GOD
Finally, let's move on from that to what happens to be my favorite transformation in the series. Super Saiyan God. As such, I think this section will sound a little more apologetic than what came before or what comes after, but I will try to keep it to a minimum.
The form was first achieved by Goku and only Goku in the Battle of Gods movie. It was achieved by a ritual, in which five pure Saiyans pour their "heart" into a sixth one. Again, the form does not break the logic of what came before it, but instead creates a new reason/method to surpass what at Super Saiyan 3 seemed to be the end.
Maybe what doesn't make me see it is the fact the two are literally opposites in every way, shape and form (heh). One has the user getting all skinny, cool and relaxed looking, while the other has veins almost popping from the intensity that's being put on the body. Not to mention the auras. Kaioken has a red crazy chaotic aura that makes Goku's whole body look red and boiling, while God's one is a lot more... tidy moving, showing that Goku is really in control of it and he is in no danger.
The original Super Saiyan changes first and foremost the color of the hair. Well, Super Saiyan God does that too. However, Super Saiyan also changes the shape of the hair, which God does not. Instead, God choses to change the shape of the eyes, making the user look a lot more youthful. Not as obvious of a change as the raised up hair, but still enough to make the features of the user look distinct.
Not much to say otherwise. The transformation is cool, the skinny build is refreshing, the magenta color does look godly and it subverts every expectation.
Wait... That's not Golden Freeza? Are you sure? It looks like it. I mean, it's literally Freeza but Golden, how can it not be?
Okay. You get the point. That is not actually Golden Freeza, at least not how he appears in the actual Ressurection of F movie, or Dragon Ball Super. It's just a quick edit I made in Paint for the whole purpose of this joke, that you probably cringed at.
This is Golden Freeza:
I mean, let's take a look at Freeza's OTHER transformations:
Then, the fourth form comes and that is a pretty big change. It is also Toriyama's best design ever made. Simply astonishing and probably the main reason why he loved doing the subverting expectations stuff so much. Because overly complicated designs are never going to nail a point on it's head the way a simplistic design does.
What doesn't make it a pallete swap? Well, from top to bottom: he gained brows, which he didn't have before, reason probably being so there could be an interupting line to color is face, which was the same color as his body previously. Also, the highlights on his shoulders are gone, instead, until his forearms, his arms are a different shade of gold. And, speaking of the forearms, the purple highlights are no longer straight, but roundish. He also seems to have gained some form of pants, also colored in the same shade of gold as the arms. The hand and feet are also now purple, whereas they were the same color as the rest of the body in his final form.
Concluding, I think the form is a work of simplistic choice that Toriyama has always done, so to complain about it for this reason is kinda unjusted, since there have been dozens before it (and a worse one in the same movie), while conveying the theme of the movie (F's revenge on saiyans) very well.