Continuing the discussion about API naming...
The BCL also has ToTitleCase, which does use the "Case" suffix, so I'd like to keep the door open for alternative names a bit.
I think BeUpperCase reads closer to regular prose.
While we should get inspiration from BCL, it does not need to dictate our choices.

How does char.IsUpper handle culture?

I recall that char.IsUpper only returns true for characters that are both letters and uppercase.
That clearly justifies that we should have all four combinations, as a character not being upper case does not imply that it is lower case, e.g. digits are neither lower case nor upper case.

I think the test suite would benefit from tests like these to reflect those cases.

subject = "A1";

Action act = () => subject.Should().BeUpper();

act.Should().Throw();

subject = "a1";

Action act = () => subject.Should().NotBeUpper();

act.Should().NotThrow();

and the complementary for [Not]BeLower.

cca1d01:

• Resolved comments
• Rebased on upstream/develop
• Updated docs