If you're really this upset about "might of," you're not ready to hear about double modals.
What do you have to say about the sentence "She might could swim?"
The difference between a language and a dialect is nothing more than an army, a navy, and money.
Your pedantic approach to language has its place inside the four walls of a classroom, but outside of a professional or academic setting, majority tends to rule - and I guarantee you you'll find a pretty convincing majority of people make the "mistake" of saying "might of" instead of "might have."
The point is that their meaning is 100% clear. To correct them is just pointless. There's nothing to gain. There's no clearer understanding to be had. No communication breakdown is going to occur because of it. There's no ambiguity to the meaning.
You're making the correction purely on your assumption that the prestige dialect is THE dialect, and your assumption that anyone who undermines the prestige dialect is somehow less intelligent. You're making the correction because you think it somehow makes you superior.... it doesn't.
I'm fully aware that the "correct" (prestige) usage is "might have."
But when I'm speaking to my friends over a few beers, or typing on a video game forum, if I say "might of" it just plain and simply doesn't matter. You're not my professor, and you're not a potential employer, and you understood full well what I was trying to say. You just chose to correct it because you think you know the "right" way its used, when in fact - there is no "right" way.
Would you go around southern states correcting everyone who says "Y'all" into saying "You all?"
Would you go around the midwest correcting everyone who says "You guys" into saying "You people?"
You say you're from Europe... don't you people say things like "Al-yoo-minnie-um" for aluminum? What's your stance on things like that? There's no "I" or "E" between the "N" and "M." Does that make many Europeans illiterate?
Everyone and their mother will know what you're talking about. Nobody would even question it. Its not the correct pronunciation, and makes no sense at all to say it that way... but its said. Its said WIDELY across English-speaking Europe. Do you go around correcting everyone you hear pronounce it that way in the same way that I've corrected you here?
I'd put money on the fact that you don't - because you know what they mean, even though its wrong.
The argument that prestige dialect is superior in some way to common language, or indicative of greater overall intelligence is asinine, man. There's no correlation. Its all about environment, and using the proper dialect for where you're at and who you're talking to.
Knowing when to be prim and proper, and when to talk however you feel like talking is the ACTUAL sign of intelligence.
If this was a professional environment or an academic setting, we "might of" used "might have."
But its not.
So get over yourself.