I've been using my mini vases quite a lot this summer, with all our wonderful dahlias for cutting. When water sits in the vase for a period of time, a film develops at the water line and it doesn't come off with normal washing.
good, easy way to remove hard water stains like this is to fill the
vase with a 50/50 combination of plain white vinegar and water and let
it sit for about 5 minutes. Then take a soft brush and scrub a little,
then rinse -- the stain is gone.
This little bottle brush is handy for cleaning vases and things you can't get a wash cloth into. I think I ordered a set of three brushes in different sizes years ago, and now I can't remember from where. The tip of this brush is a soft cottony material, and the bristles provide just enough friction without scratching. The result is a sparkling clean glass vase!