Percival P. Cassidy wrote:

For a while after we moved into this house the shower stall looked
fine, but then the grout lines -- especially those between the edge of
the floor and the bottom of the walls -- started looking dirtier and
dirtier.

The usual shower cleaning products did not do much, nor did undiluted
bleach straight from the bottle.

We thought of redoing the shower with plastic or fiberglass panels,
but it's significantly smaller than any standard off-the-shelf units.
No doubt we could get a custom liner made, but probably costing more
than we would be willing to spend.

I've just started attacking the grout with a grout saw, with the
intention of regrouting those joints, but it appears that the previous
owner used silicone caulking compound over the existing grout along
those edges. This complicates cleaning out the crevices. And even
where I've managed to clean out the old grout and caulking compound,
everything is still black, and even straight bleach seems to have no
effect.

Any suggestions about the best way to deal with this?

I regrouted our tiled shower stall with good results, about 4 yrs. ago.
I used the Dremel with special bit and guide. Only tough part for me
was kneeling for the low spots and using enough pressure to regrout (I'm
old ) The black stuff you find is bothersome, and I hope it is from
the previous caulking applied over dirty old grout and not from a leak
that has saturated the wall. If you get through all of the grout and
still find blacfk mold, I've be very concerned that there is a leak
behind the wall. Any chance the original grout was black? When you get
it cleaned out, use full strength bleach, rinse well, and dry well
before you regrout.

I ended up with a few grout joints that didn't fill entirely, which is
due to my lack of muscle and dim lighting. I'd be sure to check the
grout for flaws with light across the joints to catch the little "misses".

The smallest tip for the Dremel tool is, I believe, 1/8". We have a few
tiles that are apparently closer, and the bit skipped out of the joint
but fortunately did not scratch or chip the tile.