On a typical US NEMA 5-15 outlet, what parts are generally safe to touch when it is live?
The safe parts to touch are the plastic, of course, and the metal frame that holds the outlet into the electrical box (the yoke).  Of course, this is in a correctly wired installation where the ground wire is connected and runs all the way back to a properly terminated ground point.  The grounding would prevent you from being shocked if there was an internal fault in the outlet.

If you suspect something is wrong with the wiring or there is no grounding present, you really shouldn't touch any metal without testing it for voltage or turning off the circuit at the breaker.

Also, this may be obvious to you, but both sides of the outlet where the screws are will be energized - even the screws that are not directly connected to a wire.  Never stick your fingers or a screwdriver into the box when the power is on - it's just too easy to touch the sides of an outlet or light switch.

Addition: this isn't very common in newer residential construction, but if there are metal boxes in the wall, some outlets are grounded to the metal box, and the metal box is grounded to the rest of the system.  So, if you see an outlet in a metal box with no ground wire, it could still be grounded via the box.  Of course, the box may not be grounded at all, so don't assume anything.