Just to add to @wiretap's excellent writing, I would like to emphasize again that believing that "grounding" is a safe ESD precaution is a potentially disastrous misconception.
My most recent component damage didn't result from static discharge, it resulted from the grounding of a (supposedly) potential-free circuit connected to an insulated power supply. However, there was a *very* small capacitance between mains and output, causing mains voltage hitting and shorting two adjacent output pins of a CMOS IC that happened to touch "ground". The malfunction resulted in a newly created OR gate - if one pin got high, the other one went high also.
The cause wasn't ESD in this case, but the effect was basically the same.
Again, the only proper way is to make sure that your potential is the same as the circuit you are handling.
As I am a lazy retarded dog, I do not like chaining myself using wrist straps.
I just make sure I am always the same potential by, for example having always one arm lying on/touching blank metal parts of the computer case I am working on.
The same when I work with stuff on ESD mats.
The only thing I permanently chain to "ground" (or better called "common potential") is the soldering iron, which is connected to the ESD mat via clamp.
I'd be careful about using the electric grids' "ground"... it is often not recommendable, especially if it is a high resistance grounding.
Or even worse, particularly in the cases where there are still two-wire circuits with the wall outlets' "ground" connected with the neutral wire.
As in my home the electric installation is of the latter kind, I absolutely prefer to disconnect all direct and indirect connections to grid "ground" of valuable hardware like old computers, before I work on their innards.
Thanks again to @wiretap, @dionb, @jo22 and the others who try to inform people about the catastrophically underestimated ESD risks!