Transformer people use rules of thumb like 500 circular mils/amp.  This
    takes into account the fact that the inside of a transformer will get
    warmer than a wire in open air.

    I think a reasonable estimate for your purposes would be something like
    200 cir mils/amp.

    For example, 30 gauge wire has a diameter of 10 mils giving an area
    of 100 circular mils.  The above rule says you should be able to pass
    half an amp and not melt the wire.

It takes MUCH more current than 500mA to melt a piece of 30g wire-wrap wire.
I just used my bencchtop supply to run 3A through a piece and it didn't even
glow or discolor due to heating.

That doesn't mean the rule of thumb is invalid, though.  I'd expect the
first concern to be "excessive heating", just due to system issues. (and
there's a LOT of wire in a transformer, so even a little bit of heat/cm will
amount up.  Secondly, you'd run into "don't melt the insulation", followed
by "don't melt the solder", with actual wire-melting WAY up there on the
list of concerns.

BillW