Personally speaking, if you don't have an antistatic mat to sit the electronics on then it's a bit pointless. The idea is that the person (the potential generator of static) is equipotentially bonded to the equipment that is to be worked on thus equalizing voltage levels. It's of secondary importance that both you and the equipment are earthed. That mat serves as a go-between; you connect to the mat and the equipment is sat on the conductive mat.
Regards safety issues the earth point needs to be checked at regular intervals. If earth becomes disconnected at the socket then there is no safety issue but if earth gets disconnected downstream of the socket then there will be a capacitive connection between the supposed earth wire and the AC power.
This, once caused me a problem and although I could only feel a slight tingle, it did bust some chips when I connected my oscilloscope (which happened to be earthed correctly. I measured the voltage and it was 115Vac - exactly half of the 230Vac on the bench - this is due to the net coupling (via capacitance) to live and neutral being the same i.e. it formed a potential divider.
In short, my experiences tell me that using an antistatic wrist strap without an antistatic mat is pointless. The wrist strap has to connect to the mat and the mat has to connect to ground/earth to be properly secure against static issues.