Oscar feats that have only been achieved once

Going My Way (1944) – The same performance getting nominations in lead and supporting categories

Barry Fitzgerald was nominated for both Best Actor and Best Supporting Actor for Going My Way, winning the latter. This is obviously never happening again as the rules were changed after this occurred to make a performance only eligible in whichever category it receives more votes in. There are instances where, rules allowing, this could have been repeated, such as Kate Winslet for The Reader.

The Battle of Algiers (1966) – Being nominated in two non-consecutive years

The Battle of Algiers was nominated at the 39th Academy Awards as Italy's submission to Best International Feature Film. Two years later, after having a qualifying release, it was nominated for Best Original Screenplay. Other films up until the 70s were also nominated in two separate years due to the eligibility requirements being different between International Feature and other categories, something which is not allowed anymore, but Algiers is the only instance of it not happening in consecutive years. This would be unlikely to happen now even if there weren't rules against it, as international movies generally take a lot less time to be released proper in the States.

Young Americans (1967) – Getting a win revoked after the ceremony

Young Americans won Best Documentary for 1968, but the win was rescinded after it was discovered the film had premiered in 1967, and the win was given to the runner-up instead. This is also the only the only instance I'm aware of of a nominee being officially found ineligible after the ceremony. This is unlikely to happen again as release dates are more easy to find in the current age, although the Academy isn't always the quickest to catch things that go against their rules.

The Muppet Movie (1979) – Having four songs shortlisted

The Original Song category had a published shortlist for 1968 through 1979, and again since 2018. The Muppet Movie is the only movie to have four songs show up on the shortlist, which is now limited to three, with only two from the same songwriter being allowed. Since the song shortlist doesn't encompass that many years, it's possible this would have occurred at some other point had there been a shortlist, such as with Beauty and the Beast and The Lion King. However, I have to wonder how willing voters would be to pick so many songs from the same movie. I'd have little problem picking seven songs each from A Star Is Born '76, The Muppet Movie and Fame, but I imagine the branch would be more varied. Rules aside, the main roadblock today would be studios' unwillingness to have songs from the same film competing against each other.