Felt hats - hats are made by being stretched and manipulated using steam, so hats may become out of shape if they get wet. If the brim tilts down it is sensible to store the hat on its crown. If the hat starts to look a little tired or dirty use a bristle brush (such as a new shoe shine brush) to gently brush the felt. 

 

Sinamay and straw hats - very similar care to felt hats, Some straw stiffeners are water soluble so it is not advisable to wear a straw hat in the rain, I have seen many wedding hats with dents, these can be smoothed after a very gentle steaming.