Darren Fletcher & Michael Carrick
The way in which this could successfully work is for Carrick to go forward and use his passing, which is by far his best attribute, while Fletcher assumes a slightly more defensive role. This should be our first choice partnership for Premier League and FA Cup matches when the 4-4-2 formation is used, as it allows for more flexibility in the midfield because both players are capable of both attacking and defending at different times, depending on the circumstances of the game.
Anderson & Owen Hargreaves
This partnership would be best suited as an alternative pairing to Fletcher & Carrick in both Premier League and FA Cup matches, and could be most effective when the holding midfield role is needed to shut down an opposing player, and a box-to-box player like Anderson is needed to provide attacking support for Rooney and Hernandez.
Paul Scholes & Darron Gibson
These two should be played in League Cup matches and against "minor" Premier League teams. This allows Gibson an extra season to build up some much needed consistency in his game, and playing alongside Scholesy wouldn't do him any harm in his development as a player. It would also help us prepare for life without Scholesy if he is playing less football, although he could still be used in bigger games if the experience is needed.
Darren Fletcher, Owen Hargreaves and Park Ji-Sung
I see this as our best option for a 3-man central midfield, because it gives us a strong defensive element to the midfield with the ability still to go forward. Obviously, when the 4-3-3 is used the main focus is neutralising the opposing midfield, and attacking on the counter using the pace of our wingers. This would certainly reduce the effectiveness of just about any midfield in the world, as all 3 players are very strong defensively and have a quality work ethic. The way in which this would work best is for Park to "tag" the opposing player who poses the biggest threat going forward, precisely what he did to shut down Andrea Pirlo against AC Milan, whilst Hargreaves plays the holding midfield role with the support of Fletcher, who can go forward when necessary.
Of course, these would not be the only combinations used, however I think it would work quite well if some of these partnerships were developed, as the rest of the team, and perhaps more importantly the players involved in these partnerships, would learn to structure their game around the way these different partnerships work, meaning that the team would be better organised and less confused if they are aware of how the midfield operates with different players.