Advanced Square-1 Techniques
I've assumed you've come here because you've learned how to solve a Square-1, and you want to learn some techniques to get much faster at solving the puzzle. Well, you're in the right place! Here, I'll be going over many different techniques for you to get your average down to 15 seconds, even 10 seconds!
Learn More Algorithms!
It may seem counter-intuitive especially if you come from learning CFOP for 3x3, but learning more algorithms does get you faster at Square-1, since there are less pauses involved in the solve when you do. While I do not suggest you go ahead and memorize all of the EP algorithms right away, you definitely should know all EO and CP algorithms, since they are not that hard to memorize and execute.
See here for a list of all the EO and CP algorithms.
(I'm not sure what the formal name of this is called, but I call it this.) Have you ever had blocks of pieces that were already connected during CP that you wanted to keep during the EP step? With CP preservation, this is possible, and it is really easy to do too!
See here for a short guide on how to use CP preservation.
By now, you've probably noticed that the 8-edge method for doing cubeshape is mostly inconvenient and very slow, although very easy to remember how to do. In fact, it turns out every cubeshape can be done within 7 slice moves (that is, 7 "/" moves.) You can greatly reduce the amount of time it takes to do cubeshape by learning the scallop-kite method, which, instead of getting all 8 edges together, you get the puzzle into a shape known as scallop-kite, which is more convenient to get to for some cubeshapes.
Furthermore, advanced cubeshape also eventually entails doing optimal cubeshape, doing the shortest amount of moves to get the puzzle back into a cube. This takes time to get, but the scallop-kite method allows you to progress to this easily.
See here for a guide on doing advanced cubeshape.
Note: you should only learn these once you've learned the previous techniques.
Parity is annoying; we can all agree. There's various ways to deal with parity, and by no means are they easy to learn nor feasibly learnable in a single day. I'll go over the major ways to do parity in a solve.
One obvious way to do parity is to learn all 99 EP algorithms so you know every single case of parity that there is. Unarguably, this is the most brute force way of dealing with parity, and, for most, the most obvious way to deal with parity. The obvious downside to this is that you have to memorize A LOT of algorithms, and most of these algorithms are not that pretty.
See here for a complete list of EP algorithms.
Another common way to deal with parity errors is CP parity, in which you recognize if you have parity during the CP step, then perform an algorithm that simultaneously solves CP and fixes parity errors. The great thing about this is that is greatly reduces the amount of EP algorithms you have to learn, halving the amount to about 49. The downside to this method is that recognition can be quite tricky when trying to figure if there is parity, and may lead to slower times as a result.
This method was developed by Andrew Nelson, and notable users include him as well as Tommy Szeliga (who later switched to full EP).
See here for a list of CP parity algorithms.
The last method is known as cubeshape parity (CSP), in which you do what the name suggests: fixing parity errors during cubeshape. This method is by far the hardest method to understand and takes a great deal of practice to get right. The big advantage is that with this method, you check for parity during inspection, thereby not influencing the solving time. Furthermore, doing parity during cubeshape will, on average, save more moves than the previous two methods.
This method was developed by Matthew Sheerin, and notable users include Lakshay Modi (who was one of the first, if not THE first, to use CSP in competition), Tommy Szeliga, and Rowe Hessler. Many more people are gradually grasping this method as it takes more ground.
See here for a full guide on how to do CSP.