The small particles of dirt left on pieces of clothes which only Rose Spires, the Cheer laundress, can see and clean in combination with Cheer. In her words, dingies are:
Dingy is the opposite of clean, love. Its dull, grayish, faded, grimy, grubby, mucky, and just dreadfully dirty
Dingies is Dundee slang used to tell someone that their talking a load of bollocks.
* These videos are coming directly from Youtube, they may or may not be most relevant to the word "Dingies"