So leftists always think the world is misogynist, white supremacist, heteronormative, etc; rightists usually think it's mostly unbigoted. I'm going to lay out what I think of it and why. Currently the page only discusses gender, but I plan to add discussions of race and other traits in the future. (My analysis is also specific to America.)
I'm not going to claim there isn't misogyny in modern America. But most of it is a matter of cultural beliefs and tropes like the damsel in distress and the asymmetric treatment of nudity; I don't think there any widespread examples of actual abuse being considered okay specifically against women. And the cultural stuff matters. But I also think they're mostly things that have a directly corresponding misandrist version.
xkcd How it Works shows a subtle but telling asymmetry in the way individuals' failures are interpreted as evidence for group differences. When I first read it, I hadn't thought about it much before, but it made me realize I've heard stuff like that everywhere. I think I've even said things like it myself.
But on closer inspection, it goes both ways. That specific example happens to women because STEM fields are stereotyped as masculine. I've heard people make similar comments about men in areas that are expected to be filled by women, such as cooking or child caretaking.
The wage gap. I don't think there's any good evidence for this because a whole bunch of problems related to statistics and citations apply; in particular, the all other things the same requirement for comparisons to be meaningful is unprovable. Even if we're controlling for number of hours worked, education and preexisting experience, similarity of working conditions is wildly open to debate. You can't aggregate that in a stat.
Another reason I don't believe it, specific to the software industry, is that the male supermajority seems more exaggerated in open-source communities than in enterprise development. If women were systematically gatekept or underpaid, you'd expect there to be a higher proportion of them when you remove wages and employers with the power to gatekeep.
There are some very widespread forms of socially acceptable violent misandry though.
Compulsory military service. Men in America are required to register for the "Selective Service System" - "service" of course meaning slavery. If you don't already know, I encourage you to read about the punishment for refusing on that article.
Asymmetric violence rules. A man physically striking a woman is seen as scandalous almost regardless of the circumstances, but a woman physically striking a man just for saying something she doesn't like is often seen as funny instead of grounds for retaliation. I mentioned this in the article on sexist tropes in fiction since it occurs most often in fiction, but it happens widespread in real life.
Legal system bias: the proportion of American prisoners who are men is well over 90%. I know this doesn't constitute bigotry, but besides the sheer magnitude of the disparity, there's good reason to think this is a real indication of it: the bias pervades our culture. Ordinary people's stereotypes of criminals are always male. The proportion of our fictional villains who are male is what, 90%? (Compare that to the proportion for heroes.) Judges, cops, and the assholes who call them all come from a culture where criminality is seen as masculine, so it would actually be surprising if there were not misandry involved, even before knowing there's any disproportionality.
No doubt someone will claim it's justified because most criminals are men. I think you have no idea how circular reasoning works if you make that objection. Those stats come from a culture where convictions are decided by people who have this preconception. They're not a count of criminals, but of convicts.
It's infuriating that lots of leftist websites will actually spin the incarceration stats as misogyny. They'll say things like "the incarceration rate for women has been rising twice as fast as for men for the last two decades", ignoring that prisoners are still almost exclusively men.
Circumcision. I left this for last because honestly, it's tasteless to mention anything else after this. If you know about this and still claim men are 'privileged', that pretty much convicts you as a crazy misandrist.