American Society c. 2000
The science of this world's 20th century unfolded basically identically to our own and we can pick up in 2000 fairly easily. Due to the fear of supers and curtailment of "unusual" abilities, though, the government of the United States is considerably stronger and something to be reckoned with, as are many other governments. Furthermore, most technological advances have been more regulated and thus restrained. Citizens register their cell phones as well as any computing device and any other communications or processing devices. The internet does exist as today's does, but it's much less free-form and private sites are discouraged by the registration process. What the internet does do well is keep track of citizens and their devices. In other words, we've become very 1984 in some ways.
While this may sound like a game of Paranoia, I should point out that life isn't perceived to be bad by people. While we haven't had the massive economic growth of the last 20 years, we've had reasonable growth and a more organized society. We do a better job of feeding the poor and controlling crime. It's rumored (to the point of being accepted as fact), by the way, that registered supers even if not officially government agents are put to work as informants and even enforcers. That's probably why the streets are as safe as they are.
And what about the many non-registered mutants/mutates? They are common but all hunted by the authorities, to varying degrees. The more law-abiding/good they are, the more they tend to be left alone. Some localities even blatantly ignore benign supers, though the FBI never does if they become involved. The more destructive (even just to property) they are, the more hunted they are. "Good guy" supers have a code of ethics where they watch out for each other, helping each other to not be captured. This usually extends to the not-so-good-but-not-bad supers, those who are destructive or even deadly to super-villains. Some mutants, both good and bad, are politically active and advocate for more freedom and tolerance. Others are politically apathetic but don't register as they want a somewhat normal life. That doesn't mean they can ignore their abilities, whether for profit or for societal benefit.
An important note here to include is that the avant-guard is a vigilante group that hunts mutants with the intent to capture and turn over to the government or kill them, depending on the particular cell. It's a national organization with some international associations. They are your basic agents, with kevlar armor and (illegal) big guns. As implied, they are organized into cells that have very limited knowledge of the overall organization. They are united in their hatred and fear of mutants. They do not go after humans who do not appear to be mutants, even if those humans have extraordinary powers.
There are many theories as to why super-humans came into being. The most popular theory is ambient radiation; the nuclear tests in the deserts seemed to be linked to the earliest supers. Laboratory tests are inconclusive but have not disproven this. Another popular theory is that these people have always existed but the appearance of Superman in an era of mass communication brought it to light and encouraged other mutants to step forward. The third popular theory is that it represents a new breed of humanity - this is supported by a genetic strand so far found in over 90% (but not all) of the super-beings. The problem with this theory is that mutants tend to have such different abilities. A less credible theory is that all humans have abilities and that it's all psychological, thus explaining why Superman existed so closely to the comic book version. That's disputed by most scientists. The "crackpot" theories attribute mutants to UFOs/aliens, government experiments/world conspiracies, and the supernatural.