This isn't perfectly answerable, because it's easy to come up with specific things that only occur once if you get complicated enough.
However, if you limit it to things that would merit a sports news organization writing an article about it, then probably the most rare occurrence is a team scoring in every inning for a full nine inning game (meaning, they have score in 9 innings out of 9 innings; this mostly specifically excludes home teams since they'd probably be winning and thus skip the ninth inning).
As of this article's publication, it had only occurred twice in modern history: 1923 and 1964 (and 4 more times overall, pre-1900). Since then it's happened one more time, in 1999, in a game I watched on television (COL@CHC; back in the day when COL had some really good bats and CHC was, well, your father's cubbies). The earlier article does a good job explaining how likely this is to happen - once every 250k games, more or less, which is about once every century if you assume every inning is an independent event (protip: they aren't). It's happened three times in the last century, which isn't that far off I suppose regardless.