Caps were worn by all classes, the difference being primarily the fineness of the fabric and level of decoration with ruffles and other trim. Women weren’t as likely to wear caps in a formal setting as in doing work or “undress” (at home). Almost always, the caps were white, the exception being a pink cap in one portrait. They might be cotton, linen, organza, etc. Silk ribbons, in any number of colors, were often used to ornament caps. Caps were worn over dressed hair and sometimes a hat was worn over the cap. Today, the caps are de rigor for covering a modern hair style.
What you won’t find in searching period images is a “mob cap” which looks like a round shower cap. Caps from the period have a distinct shape created using three main pieces – a caul (the body), the band, and ruffles. A cap can have a single ruffle or be double ruffled. It might also have a caul ruffle and another ruffle framing the face.
If you have good sewing skills the American Duchess shows you how to make your own cap at: See: https://blog.americanduchess.com/2017/12/correction-1780s-cap-pattern-american.html or Kannik Korner and others offer patterns which I prefer. Generally a stiffer fabric makes up nicer and is less likely to collapse in wearing it.