Stinking water from ONE faucet
Need more information.  Sorry for doing this as an answer but the alternative was a raft of comments.

Can't speak for your location.  Here we can get a chemical water sample bottle from our local public health office, fill it, and send it in.  A month later we get a report.

If you can do this with the affected faucet, you can find out WHAT is generating the stink.  Hydrogen sulfide (rotten egg smell)  Sulfur dioxide (burned match smell)  Dead animal (rotting meat) smell?



How long does it take for the smell to regenerate?  E.g.  You ran it until the smell stopped at 8:00.  Does it stink again by 9?  



Where does this line run relative to other lines?  Is it the last tap on a run or is it a side tap?



Are your water lines copper or plastic?



Do you have water hammer traps?  (A capped vertical pipe branch.  They tend to collect dissolved air, and act to reduce the Clunk!!! when someone shuts a tap abruptly.  A dead mouse in one could add a certain bouquet to the water.)