Finding the right synonym for "most occasionally" can be tough, as the phrase typically refers to something that happens infrequently or rarely. However, there are a few words that could work depending on context. "Seldom" and "rarely" are two options that convey a sense of rarity or infrequency. "Intermittently" suggests that something happens at irregular intervals, while "sporadically" implies occasional occurrences without a regular pattern. "Infrequently" is a more straightforward option that simply means "not happening often." Ultimately, the right synonym will depend on the context and intended meaning of the phrase in question.