I hadn't heard anything like Two Door Cinema Club until I heard them.

Justin Bieber is the first that comes to mind. Believe showed how much he has grown, and that he is able to break away from the mold. A unique sound and a beautiful message.

This.

Muse are the ones that came to mind as they do like to experiment a lot. Not a big fan of there last album (still had some fantastic songs though) and I'm undecided on this new one, but I never get bored of listening to Origin of Symmetry. They really do try to mix a range of styles in their music though.

Otherwise, maybe Enter Shikari? They also mix multiple styles to come up with their own sound.

Linkin Park have always been pioneers and their new stuff is always changing. But I like your pick with Mumford and Sons too. Also Dropkick Murphys.

Muse was the first to come to mind. I remember reading that they were all open-minded on the music which is why they like to experiment. They created some cool &somewhat unique stuff like Exogenesis, Knights of Cydonia and many of the songs from Origin of Symmetry.

Tom Waits.

Imagine Dragons, my current "band of the moment".

They're not all that original when it comes to lyrics, but their sound is quite diversified and pleasant.

Sure, they do alternative indie rock, yet I feel they manage to go above the standard fare.

M.I.A. - original and has an impact on directions taken in the mainstream. I don't know if any other act has quite that impact.

Lady Gaga has most easily had the biggest impact on the direction of mainstream music in recent years.

Katy Perry is not actually a song writer; other people do her music for her (and are copying existing pop styles that are big at the moment), she just performs.

Regina Spektor.  She does things her way, not following the typical "rules" and her sound is quite awesome!