short answer: h
i don't feel like typing the long answer.
breaking the back rank mate
Sort:
Depends on the position :p
g and h more common than f, especially with queen on the board. Other than king safety (which diagonals are more vulnerable, which color complex is more vulnerable, possible pawn breaks and sacrifices to remove pawn cover) moving the f pawn also loosens control over e3 (or e6). Sometimes worth mentioning...
Another king safety issue is along the rank. With f pawn up there's less cover for the king on the 2nd rank.
With queens off there are other endgame considerations. How fast can your king make it to the center? g pawn lets you start moving up ranks right away while making you wait on files. h pawn you start with files but have to wait a move on ranks. Of course it always depends but I may venture a guess and say ranks are more important so start with the g pawn when that's an issue. Of course f pawn lets you do both at once (diagonal move!)
Also affects pawn structure i.e. where the base of the pawn chain (the weak targets) are. With g3 h4 the weak link is closer to the center and sometimes that means it's easier to guard. With g3 f4 it's on h2 and sometimes that means it's harder to attack.
So king safety (and later activity) and pawn structure are the main concerns. The squares you leave behind, probably most notably e3 and e4 I'd say, are sometimes a consideration too.
In general is the f,g, or h pawn preferrable. I know there are a lot of ifs. But particularily with queens on board I know the f7,g7,h7 pawn setup is pretty strong. To free that back rook, is one of these pawns preferrable to move first.