Is the neutral wire considered safe?

The neutral is NOT safe to touch.

When everything is working correctly, it should be at most a few volts from ground. However, and this is the big gotcha, if there is a break in the neutral line between where you are and where it is connected back to ground, it can be driven to the full line voltage. Basically in that case you are connected to the hot line via any appliances that happen to be on in that part of the circuit. Those can easily pass the few mA it takes to kill you.

This is not supposed to happen, but since failures can be lethal and costly, a extra layer of safety is built into the protocol and rules. It is irresponsible and needlessly risky to consider the neutral line safe.

This is why modern appliances either have two prongs and everything is insulated from the user, or three prongs and anything conductive the user can touch is connected to ground. In some past cases there have been appliances with polarized 2-prong plugs, but those are seriously frowned upon today. I don't think you're going to get UL approval for such a device unless it is fully insulated, in which case you shouldn't need a polarized plug.

On a non degraded correctly wired installation the neutral wire is safe because it is at the same potential than the ground terminal.

It is true that it carries current but because there is no voltage difference from ground there is no current passing through when you touch it.