This is a list of every film which received an Oscar in at least one of the following categories:
Best Supporting Actor
Best Supporting Actress
No film has won Oscars in more than four of these six categories. Fourteen films are tied at four Oscars. I should note that one of them - 1934's It Happened One Night - won four before there were categories for Supporting Actor and Supporting Actress.
Which Oscars a film received are listed in the notes.