Is a single wrist strap enough against ESD?
Grounding yourself is only part of mitigating ESD. You can still cause damaging discharges while properly grounded when the device you're working on is itself charged, when the tools you use (like a multimeter or screwdriver) are carrying a charge or when your workbench/parts bin is charged.

Ideally you'd ground everything that can come into contact with sensitive devices:  


Yourself  
Any surfaces where you may lay down your board and/or components 
Any tools and equipment that may contact the device (soldering irons, electronic test equipment, screwdrivers, etc.)  


This grounding can be accomplished either through your body (as is done with a dissipative screwdriver, side cutters and the like) or a dedicated grounding wire (e.g. an antistatic mat on your bench). Grounding leads must contain a high value resistor, so that you won't get shocked via your strap when accidentally coming to contact with high voltages. Grounded surfaces should be resistive (e.g. ESD mat, dissipative tray), instead of highly conductive (e.g. steel box, aluminum plate), so that any discharges that do occur will get current limited.



Do you really have to go through all this trouble? It depends. I personally don't bother taking all these precautions with hobby electronics or while taking my PC to bits (at my own risk). However, I do take this seriously at the workplace and I think you should do the same.