Bonnets

While the bonnet is considered the hat of the early 19th century, it has some very distinct roots earlier on.  I found images starting in 1772, consistently appearing until the turn of the century.  A single image from 1740 also appears to show a very bonnet-like hat, a bit different than it's later cousins, but certainly not the plain straw hat often seen.  Another from 1765 is much more like the images below.  There is a much larger variety of sizes, shapes, and trims than I expected to find.  Black does seem to be the most common, but white is also very popular with the more fashionable sort.  Although I've seen pictures of brown, blue, pink, lime green, and even straw!  The basic form seems to be universally worn by all classes.
The biggest problem, however, is the distinct lack of extant examples.  I found only two, very different in style.  I wouldn't be surprised if there were more out there improperly dated to the 19th century, but fortunately the bonnet's popularity is quickly increasing and there may be better research to be had upon it in the near future.

Old Macaroni Miss-led, 1772

A Ladie's Maid Purchasing a Leek, 1772

An Evenings Invitation, 1773

The Invitation, 1774

A Call to the Unconverted, 1774

Is this my Daughter Ann, 1774

Twelve Fashionable Head-dresses of 1775

The Young Mendicant, 1776

The Lover's Disguise, 1776

Miss Tipapin, 1779

The Liberty of the Subject, 1779

A Lady in Waiting, 1780

The Bird of Paradise, 1781

Ensign Rosebud, 1782

Reapers, George Stubbs 1785 

 Haymakers, George Stubbs 1785

 The Fortune Teller, 1789

Extants
Black silk hat, dated late 18th-early 19th, appears very much like Ensign Rosebud or The Invitation
Black silk hat, 1770-80

Pennsylvania Gazette Index of Bonnets
Two other museum images here and here.