Using 120V AC through a Breadboard (good or terrible idea?)

"Double insulated" - the requirement for not having an earth wire - also means that the transformer itself must have two defined layers of insulation between primary and secondary (and that does not include the enamel coating on the wire). If this is the case, there is no need to ground the secondary in any way.

rclark43:
Perry, I'm still a little confused. If the neutral becomes disconnected then how is the power transmitted? Doesn't the circuit break open? Wouldn't any power stored between the live (hot) wire and the load be forced to ground from that? Perhaps either through a ground cable, or something else?

If the neutral becomes disconnected, then any and all appliances connected to that neutral will form a connection to the live. Power will not be transmitted because the circuit has been broken, it will be obvious that something is wrong, but until action is taken, those non-functioning appliances will be holding the neutral to live and contact with the neutral will be dangerous.