Anyone familiar with Belgium's great reputation in the field won't be surprised when I say that, while searching for textile museums in that country, I found countless references to the finest bobbin laces and the most sumptuous tapestries.  Not to imply that the results of the search were predictable to the point of becoming boring, but here are three pages with unexpected subjects, which would have made the search worthwhile on their own:  

• a picture of what Victoria Beckham's iron would look like if La Posh had been born in Rubens' times;
• a permanent exhibition dedicated -not to the culture of flax, cotton or any other textile fibre- but to the fuller's thistle. In Spanish it is easier to spot that the verb to card is related to the thistle (cardo, from Latin), but I never gave much thought to how the fullers used their thistle for carding or where all those thistles came from. The Petit Musée de Fleron promises to answer all these questions. (Google Books also provides a thorough explanation, in Spanish: Encyclopedia metódica. Fábricas, artes y oficios, 1794);
• my favourite museum from the Belgian list, the one that I have promised myself I will visit next year: a garden dedicated exclusively to the plants used for dyeing fabric in all four corners of the world, with a workshop where the visitors can try the dyes for themselves.