Personally I'm annoyed with the whole anime trend simply because it seems to be a lot of the same material, same style, etc.. etc.. being drawn over and over again. There's no variety and while I do happen to like anime, but I'm personally annoyed with is the whole let's make everything into an anime. It's being done with american animation and it annoys the heck out of me. If it's good anime inspired work such as avatar I don't mind, but don't even get me started on teen titans.
The whole net must draw everything anime also annoys me as well because I've known so many people who refuse to consider that anime itself is not the here all, be all of all work and that it in and of itself has a lot of flaws.
Plus, learning new styles is good any day of the week anyway.
As for the other, fanart. I like fanart if the person does their own twist on it, but what really annoys me is when people take an image from a manga or an anime, still it and then copy it. It's good for getting anatomy and style down initially and I don't mind it necessarily if an artist is just starting it. What I'm personally annoyed with is when skilled artists decide to do that and call it art when what they're doing has nothing to do with taking that character and providing their own version of it so much as plagiarizing that character and then calling it fanart.
And lastly, the last thing I'm annoyed with..
one thing current American animation. The flat style that is being pandered around in shows such as American Jake Long, and Foster whatever.. it's called annoy me as well. They're flat and there's no depth to the characters at all. It's not drawing to me and honestly to me it often looks like a 5 year old could draw them, i.e. ed, edd and eddy anyone.
The animation for that show was horrible and I refuse to watch a show with bad animation in it.