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. Additionally, all three of Bucky's younger siblings are girls, with the oldest one being named Rebecca.
Following Captain America: The Winter Soldier, it is generally believed that Bucky has suffered brain damage from the chair and the cryofreezing, though the extent of it and what forms it takes varies.
Some people believe that SHIELD was named such by Howard Stark, in honour of Steve Rogers. It's made very clear that Howard considers not finding Steve his greatest failure, and he's also one of the founders of the organisation.
Scarlet Witch is interpreted as being Jewish, just like her comic book counterpart.
Clint Barton cannot function without caffeine and/or pizza.
Despite having no screen time together, Peter Parker and Shuri are often depicted in fan made content as having a close friendship if they ever were to cross paths — they’re the same age, have a common passion for science and discovery, and have similar, playful personalities.