If I lived in a hard water area, I'd use a little diluted vinegar before the final rinse myself, but I'm in a soft water place.
You can purchase small ball bearings covered in copper, which are soft-ish but heavy and can be used to swirl around the bottoms of difficult places. They're called "Magic Balls".
And if you time your trip to the ironmongery bit properly, you can embarrass a young man by asking him if he has ...

I once ruined the rim of a vase trying to clean and dry the bottom of it, the rim had been cut and polished, it came to a really fine point on the inside, and I took chunks out of it by wiggling a bit of kitchen roll around on the end of a knife - I levered bits off from the fine rim.

I'm now a lot more careful and have learned a few good tips from folk here.
