TyeDye
Regular
I found this method just searching around the wiki a few days ago and I thought that it was very interesting. It was created by Cary Huang (Cube Roll on YouTube) and he made a video explanation of it as well here:
It is pretty much based on the fact that when a square-1 is not in cubeshape, you can control parity easier based on the fact that parity occurs when a square-1 is brought out of cubeshape.
Steps:
5 or 6.) Permute both layers of the puzzle
I really enjoy this new method of square-1 and I would like to hear thoughts from others on it as well 
Edit: Forgot to add an example solve.
(4,-4) / (-3,3) / (0,6) / (-1,6) / (2,-4) / (-4,6) / (-2,-1) / (0,-4) / (0,-1) / (6,0) / (1,-1) / (-5,6) / (-2,-1) / (0,6) / (-2,-5) /
(2,-4)/(0,1)/(-1,0)/(-3,0)/ Cubeshape (Step 1)
(-5,6)/(-1,0) Edge Orientation (Step 2)
/(-3,-3)/ Barrel/Barrel (Step 3)
Since there are no already paired edges and corners, I just chose the UL edge pair to start with
(2,-1)/(-4,-4)/(-4,0)/(0,6) First Quadruplet put on bottom left (Step 4a)
/(-4,0)/(2,0)/(-2,0)/(2,0)/(4,0)/ Second Quadruplet on bottom right (Step 4a)
Now I use the 2-look algs for Step 4b (I used regular notation to keep things consistent but it's much easier on the eyes using U and D notation.)
(-4,0)/(-2,2)/(2,0)/(0,-2)/(0,2)/(0,-2)/ First Alg
(2,0)/(-2,0)/(4,0)/(-4,0)/(2,0)/(-4,0)/(2,0)/(-4,0)/(4,0)/(-2,0)/ Second Alg
I do permutation first
(-5,-3)/(3,-3)/(-3,-3)/(0,-3)/(-3,-3)/(0,-3)/(-3,-3)/ Permutation
(3,3)/(-3,-3)/ Cubeshape and finish
Steps:
1.) Cubeshape - Yeah, I know I just said that it doesn't use cubeshape but it only needs it for a second. Honestly, you might be able to bypass it by grouping the edges a certain way but for now, cubeshape is the way to go.
2.) Edge Orientation - Just worrying about the edges, get all of your top and bottom colors on their respective faces together.
3.) Barrel/Barrel - Go back to barrel/barrel shape w/ either /(3,3)/ or /(-3,-3)/ From here, your edges should be grouped in four pairs with opposite colors together in the same orientation on all pairs (yellow edge on left, white edge on right). From now on you have to think of your edge pair as one piece. Do not split any of the edge pairs up as this will mess it up.
4.) Quadruplets - This is the meat of the solve. You are going to solve what is called a "quadruplet" of pieces. It is two corners and two edges (one edge pair). It should look like half of a barrel shape split symmetrically.
a.) Solve two quadruplets intuitively - Without breaking up your edge pairs, you want to match the corners up to both of the edges on one of your edge pairs. This will create a corner-edge pair-corner "quadruplet". This is generally done intuitively. Same way that you started out with square-1 and when you were learning cubeshape, they told you to just mess around with it until you got familiar with it, that's pretty much what you're gonna have to do here. Once you have one quadruplet, you can put it on the bottom left and create another on the top using a second edge pair. The way that I handle this is by treating the bottom as bandaged where I can't move the first solved quadruplet on the bottom left and only using slice moves and the U layer, I pair the corners. This also takes some practice but after a while is very easy and fast. Once you have your second quadruplet, place it on the bottom layer across from your other quadruplet. Your bottom layer should be in a barrel shape.
b.) Solve the third and fourth quadruplets - This is where the algorithms come in. In the video, he uses one swap algorithm to swap two pieces in the top left. The notation for yoyleberry is different as an edge pair is treated as one piece. With this in mind, we can use U, D, and slice moves to show the algorithms. The algorithm is this / U D' / U2' / D2' / U' / D2' / D' / D / . However, a 2-look algorithm set was proposed by Cale Schoon in the comments of the original video. http://bit.do/sq1noparity This splits the last two quadruplets up into turning the top layer into a barrel shape with similar colored corners across from each other (if it is not already) and then permutes the corners into their correct positions. The first step is fairly easy to understand and execute but the second step does have some things that make it a little confusing. With practice, it gets easier to understand the second step.
5 or 6.) Turn the cube back into cube shapeb.) Solve the third and fourth quadruplets - This is where the algorithms come in. In the video, he uses one swap algorithm to swap two pieces in the top left. The notation for yoyleberry is different as an edge pair is treated as one piece. With this in mind, we can use U, D, and slice moves to show the algorithms. The algorithm is this / U D' / U2' / D2' / U' / D2' / D' / D / . However, a 2-look algorithm set was proposed by Cale Schoon in the comments of the original video. http://bit.do/sq1noparity This splits the last two quadruplets up into turning the top layer into a barrel shape with similar colored corners across from each other (if it is not already) and then permutes the corners into their correct positions. The first step is fairly easy to understand and execute but the second step does have some things that make it a little confusing. With practice, it gets easier to understand the second step.
5 or 6.) Permute both layers of the puzzle
I perform this step before turning the cube back into cubeshape. I've recognized that when you turn a solved cube into a barrel/barrel shape, it has the top two quadruplets with the same color along the sides and the bottom two quadruplets with opposite colors along the sides. I use this information to permute the layers with the square-1 algs from Sarah's website: link. Then I turn it back into cubeshape, in which it is solved (possibly with the middle layer misaligned but that's an easy fix)
Edit: Forgot to add an example solve.
(4,-4) / (-3,3) / (0,6) / (-1,6) / (2,-4) / (-4,6) / (-2,-1) / (0,-4) / (0,-1) / (6,0) / (1,-1) / (-5,6) / (-2,-1) / (0,6) / (-2,-5) /
(2,-4)/(0,1)/(-1,0)/(-3,0)/ Cubeshape (Step 1)
(-5,6)/(-1,0) Edge Orientation (Step 2)
/(-3,-3)/ Barrel/Barrel (Step 3)
Since there are no already paired edges and corners, I just chose the UL edge pair to start with
(2,-1)/(-4,-4)/(-4,0)/(0,6) First Quadruplet put on bottom left (Step 4a)
/(-4,0)/(2,0)/(-2,0)/(2,0)/(4,0)/ Second Quadruplet on bottom right (Step 4a)
Now I use the 2-look algs for Step 4b (I used regular notation to keep things consistent but it's much easier on the eyes using U and D notation.)
(-4,0)/(-2,2)/(2,0)/(0,-2)/(0,2)/(0,-2)/ First Alg
(2,0)/(-2,0)/(4,0)/(-4,0)/(2,0)/(-4,0)/(2,0)/(-4,0)/(4,0)/(-2,0)/ Second Alg
I do permutation first
(-5,-3)/(3,-3)/(-3,-3)/(0,-3)/(-3,-3)/(0,-3)/(-3,-3)/ Permutation
(3,3)/(-3,-3)/ Cubeshape and finish
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