Apart from Singer f**king up the X-Men continuity that was set up for him by Mathew Vaughn, the problem with X-men, imo, is 16 years after the first movie we're still dealing with the same main characters.
Huh? Singer's been involved in all of the main title movies except for Last Stand. He's got storyline and concept credits for all of them.
As for the characters, I think Magneto and Mystique need to take a step back but the teams themselves have had a fairly good variety of the core characters themselves.
The next movie looks like it's setting up Mister Sinister as the main villain and there's some immediately obvious avenues for introducing new characters. Mister Sinister essentially is the "creator" of Cable. Some X-Force characters have already been introduced like Deadpool and Bishop but there's also a few people we haven't seen yet like Cannonball, Domino, Shatterstar, Moonstar and more. An X-Force movie could have a more darker tone to the established franchise, perhaps even in the MA/R rated space given the content of the original comic series.
Other than that I'm not sure there are a lot of characters left that haven't already been introduced in some way already. Longshot, Dazzler, Forge, Legion and Magik are the only real notable ones.
I'm fully anticipating that when Hugh Jackman hangs up the Wolverine boots the character will be replaced by X 23.