Which of the pawns in front of the king should be moved in order to prevent a back-rank mate? Assuming we're playing white and a bank rank mate is being threatened. Which one of the pawns (f, g, or h), in most cases, should be moved? Is there a general rule for this?
Of course, the concrete position is always more important than general &quot;rules&quot;. However, as some guidelines of what is typically played:

f2-f3: Rarely. It opens up a dangerous diagonal (a7-g1) and usually also the second rank (normally, the d- and e- pawns were moved early on). Even if Black can't mate directly, they can often at least kick White's king around.
f2-f4: Occasionally, to immediately counter-attack a Black center position on e5 (if the second rank can be covered through other means).
g2-g3: Occasionally. Also opens up a diagonal (a8-h1), but that can be okay if Black has no queen and light-squared bishop anymore to exploit it and/or if White can play their own light-squared bishop to this diagonal.
g2-g4 or h2-h4: Occasionally. Can make sense to grab space when there are few pieces left (the pawns get vulnerable to piece attacks) or sometimes even to launch a counter-attack on Black's king.
h2-h3: Statistically, probably the most played among the pawn moves. It creates the  smallest weaknesses on paper - but as can be seen from the many alternatives, it's by far not a move that should be played in all positions. Especially not if Black controls h2 with another piece, obviously.
No pawn move at all: The better the player, the more likely they are to try and parry the threat without actually moving pawns. Usually, moving flank pawns to open up a vital escape square for your king doesn't do anything else on the board (especially when moving to only the third rank) and can thus be seen as a waste of tempo (on top of creating structural weaknesses). Sometimes, that's a sacrifice you just have to make to prevent mate, but if there is another way (covering the checking square with a piece, blocking the way, etc.), it's often to prefer.