A small correction: although electronic tuners tend to show only sharp names, not flat ones, the open strings of fleas tunings are normally considered to play sixth chords, and thus the tuning F A# D G is better known as Bb tuning: F Bb D G (= Bb6). Bb is by far the more common spelling for that pitch—of course, the correct spelling always depends on context. In the case of tuning, you're talking about dropping the common C tuning by a whole step, and thus, as with the other three tuning pitches, the spelling should drop by one letter, not two, so Bb is the more correct spelling.
If the tuning were based on A#6, the spelling would be E# A# Cx Fx (x represents a double-sharp)—yucko! The key of A#, if it existed in standard notation, which it doesn't, would have 11 sharps: three sharp notes, four double-sharps. A#6 is a valid chord—usually occurring as a passing chord—but you'll rarely encounter it in the popular keys, unlike Bb6, which you often see in keys like Bb and F.
I mention this because I see Bb tuning misspelled so often, primarily due to the sharp bias of tuners. I regret if people think I'm making a mountain of a molehill; I'm just trying to clarify why it's Bb instead of A# and why it matters.