Insulation – Does Fiberglass Insulation Grow Mold in Florida?

Greenboard is sheetrock where the paper is water resistant (but not necessarily water proof). A better alternative to greenboard is to use a paperless sheetrock product like Densarmor, which uses a fiberglass backing rather than paper (it's the paper that mold likes).

Note that these products aren't designed for wet applications, however. At best, they are designed for areas that 'are sometimes damp'. If you have mold issues under your sink, the problem is water. You have a leak somewhere. That has to be fixed. Replacing the sheetrock--even with one of these other products is just a bandaid and the problem will return if the leak isn't fixed.

Vapour barrier goes on the cold side of the insulation.

The problem here is that for an unheated, Un-airconditioned shed, the outside will always be the cool side.

Insulation won't stop heat transfer, it only slows it down. A closed shed, no matter how well insulated, will always turn into a sauna.

Now you have a hot, steamy inside, and a (relatively) cool outside (especially at night). If you put the vapour barrier on the inside, sandwiching the insulation in, the condensation will occur within your insulation. Hence, the mold issue.

You're better off without a vapour barrier in this situation. This way the batts can dry out. Even better would be some rigid foam insulation, but I really think that insulation in this situation is a waste of money.