Connected to the above, popular theories arise concerning Hawkeye's backstory, many inspired by the mainstream comics. The most common are Clint was a Circus Brat and/or has a criminal record (which again, is basically his comics backstory), formerly served in the US Army or in some kind of Law Enforcement division, is a single father with a young child, is dating/married or was formerly dating/married to Bobbi Morse, that Bobbi is the mother of the aforementioned child, and many more.
Captain America: The First Avenger, having changed Bucky Barnes from Cap's boy sidekick to his childhood friend of the same approximate age left a huge window of opportunity for fans to fill in the gaps of Movie!Bucky's pre-war history. Popular backstories include Bucky and Steve growing up in the same orphanage, Bucky working at the docks before he was drafted, and going to dance halls with pretty dames being one of his hobbies.
Movie Night — The Avengers are likely to have one of these in whatever fanfic you read.
All of the Avengers will be living together in Stark Tower, either full-time or just whenever they need a place to crash. Bruce and Steve, by virtue of having no other friends, are almost always full-timers.
In a sub-universe of this: Tony is a new teacher at a school/buys a bakery and gets the other team members in. Commonly results in Loki as a Sixth Ranger and/or cameos from Darcy and Peter.