From what I've seen, published fiction uses just one type of quotation marks, but many fanfics use both: double quotes for speech, and single for describing labels, signs, or even inner thoughts. For example:

I'm wondering if this is something to be embraced, and if there's any kind of "official" endorsement for it. On the surface, it looks good to me, because there's no way for the reader to confuse speech with other stuff.

However, I don't think I've ever seen a book do things this way. They either used double quotes for everything, or a colon and/or italics for signs (like the note in my example). I haven't read that many books in English though, and my language has different punctuation rules.

 

The Oxford Guide to Style has good style guide tips for UK English, and of course, it completely doesn't answer your question. But here's how quotation marks differ between UK and American English:

Personally, (and because I am a traitor to queen and country) I like the American version more. The double quotes just look meatier. And what can I say, I like that in a sentence.

After the first and most obvious question is out of the way of 'which quotation mark do you use for speech?', English grammar devolves into a mess of tiny rules and massive disagreements. A lot of differences are regional, so what's correct or generally accepted in one country is heresy in the next.

There's usually some mildly interesting history behind why, as well. Should you put your punctuation inside or outside your quotes? The reason we started putting it outside the quotes was probably all down to the printing press, where small characters like punctuation would often fall off or break on the end of the strip of metal, so they'd put it inside the quote instead. When printing technology got better Britain switched back to the original style and America never did.

So you can see how these kinds of medium shifts and regional shifts drive grammar rules: English doesn't have rules so much as formalised disagreements, but at least you could rely on your region and medium to tell you what to do. And then, along came the internet. We're now in an era where regions aren't so far apart, and exploring a whole new medium to boot. A whole lot of what's good practice in books would suck to read on a webpage - single-spaced paragraphs relying on indentation to be distinguished, for example - post a story like that and no one will read it.

So back on track: is fanfiction's use of quotations wrong? Yes and no, because there aren't any hard and fast rules on what to use where in the first place, as far as I can tell. But I think you're right - it really is evolving, and it's actually not even a place we haven't been before.

The grammar rules in fanfiction we're discussing here seem to be morphing into something similar: double quotes for verbatim speech, single quotes for everything else.

I'd say the takeaway from all of this is this: when it comes to the English language, you can largely do what you want. As long as you're internally consistent, you're mostly correct.