Q: What is different in any day, someday, one day? I'm wondering about the differences between in any day, someday, one day
Are they perfectly synonymous? OR they have some differences?

A: "Any day" has connotations that the day is expected imminently, especially when used in the idiomatic form "any day now".
"Someday" implies that the day is expected to be some distance away. (It is often even implied that the day may not actually come at all, and that the day is the subject of fervent (though perhaps misplaced) hope.)
"One day" does not come up very often in this idiom, but it would be similar to "someday".
