Contrary to what you might hear there are actually only three distinct shooting methods, and these should not be confused with shooting style. Shooting style is the personal interpretation of method, not a method in itself. There are many styles, some good and some not so good, but all are based on the three recognised methods. Thus Stanbury and Churchill are really styles rather than methods, despite their apparent technical differences. Total Shooting is neither a style nor a method of shooting, but a method of coaching that can be employed regardless of method.
This method entails starting the muzzles of the gun moving behind the target and on its flight line, then catching and passing it with the gun slightly accelerating throughout the shot. The trigger is pulled just after the target is passed. This is the most natural method for most people. For it to work properly the shooter must employ a moving gun mount, and be careful of the relationship between muzzle and target as the shot begins. Starting behind but not too far behind the target is a crucial factor. Mounting the gun 'dead', perhaps on the point where the target originated, and then beginning the swing inevitably means starting too far behind the target. This is a common problem with this method.
This is the only method possible when shooting the Trap disciplines, and is arguably the system used most in all forms of shotgun shooting.
A very natural method, it's ideal for the beginner as well as the expert.
The method will still work reasonably well even with poor style and technique. This might seem like a plus, but in fact many shooters never improve despite plenty of shooting since they are just practicing their mistakes.
Point and Swing
This is really a variation of the Swing-Through method. It bypasses the initial approach from behind the target, requiring instead that the shooter points the muzzles directly at the target as the gun is being mounted. Once it is fully mounted the gun is accelerated ahead of and away from the target and the shot is fired.
This method forces a combined gun mount and swing and avoids the 'mount then swing' mistake so common with Swing-Through.
Not a natural method, and can lead to over awareness of the muzzles. Also it might be considered too deliberate for same tastes.
This is an entirely different method to the previous two in that the muzzles start ahead of the target rather than on it or behind it. This method is ideal for fast close targets such as those encountered at Olympic Skeet, and some Sporting targets. Because it demands a sound technique if it is to work at all it is not a good method for the beginner.
It's perfect for fast targets where there is little time. This is why it's so popular for Olympic Skeet.
It absolutely cannot be employed for Trap targets. Also the method doesn't work for everyone, especially those who claim to see no forward allowance when employing either of the alternative methods. As mentioned, it is arguably too demanding on technique to be used by beginners.
Which of the three is the best method? That depends on many factors, not least being the man behind the method rather than the method itself. As with all things, it's important to learn good technique from the outset.