What MOS has the most duty station options?

I would say 42A. They have positions in joint assignments plus, literally every battalion size element and larger.

Probably JAG, Finance, Admin, Medic, Mechanic, Supply, and Cook.

Quick answer 42A and 92Y. Admin and supply are requirements no matter what kind of unit.

probably go with admin, every unit has at least an admin clerk.

There used to be a chart that had the MOSs by branch and the largest branch was OD, followed by QM. Suspect it is one of those two

I heard tell it was 35F. Supposedly every outpost everywhere has a 35F. I don't think that is quite accurate, but it is probably close.

Medics and Signal

From a Marine point of view, anything administrative. They are needed in virtually every unit. I would guess that would be true of the other services also.

Admin. Always admin somewhere