Is plugging a wrist strap in enough to prevent ESD?
I've been told by a few people that if you plug in a PC and switch off
  its power supply that it is enough to prevent ESD.


No, it's not enough to switch off the supply. For most PC's the working voltage has a maximum of 12V, using the Human Body Model (HBM) for Electro Static Discharges (ESD) ranges from 2kV to 8kV, which is hundreds of times more voltage. Whether a circuit is powered on or off will in most cases have very little effect on where the current from an ESD event flows. 

The point of preventing ESD is not to 'float' a circuit and remove it from ground (which doesn't happen when a PC is turned off, it should always be grounded when proper electrical codes are followed), it is to stop the charge from accumulating in the first place. 

There are transistors (mosfets) that you can destroy simply by waving your hand over them. most of them are older, and many newer electrical components have ESD protection in their inputs, but that gives you an idea of how easy it is to kill electronic parts with static electric fields. 

The way to prevent ESD is to strap in, with a 1MΩ resistor on the strap to prevent electrocution. Other ways are, wear an ESD smock, use an ESD mat and to keep humidity high-ish (like 60%) as water vapor increases charge transport through air. 

Another thing to keep in mind is that some materials like plastic and paper and clothing readily create\buildup charge so only use ESD compatible materials near sensitive parts.