When you need it (isn't that helpful). Yes, model weight matters.
Folks seem to think 3/32" is OK for a 500 square-inch model; I would only use 1/8" on a 600 squares or bigger model. People concerned about making things rigid will make the change to a bigger wire sooner; people who want to tweak flaps by bending the wire may hold off, or may use something other than music wire.
Sorry for not giving you definitive answers -- you'll find that there's a lot of personal preference involved.