I would really suggest just using 0000 steel wool. You can use Brasso to accelerate the process, but it is not necessary.
The black spots are most likely locations where the low grade metal used under the chrome begins to out-gas and it forms bumps (like little zits) that literally pop and expose the metal underneath which rusts. these black spots eventually act as conduits for more moisture and then you see lifting or flaking, then it will progress to bubbling when the rust works to below the original steel surface
The grooves should not be an issue for the steel wool. (Pic's would improve your odds of getting the best advice BTW)