I don't think there was a national style of the 19th century. I think that there were many different styles being used throughout the 19th century and to say that one was dominant over all of the others is hard because of the fact that there were so many different styles being used. First was the neoclassical which included Greek revival, rationalist, idealist, and federalist. With the Declaration of Independence being signed in 1776, there was a need to show America's Independence through the use of Federal Architecture. This architecture could be characterized as decorative but sophisticated.

The 19th century was based off of revivalism. Mainly Greek and Roman. Specific styles that were used the most in the 19th century were the Beaux-Arts Style and the Greek Revival. Another style that was used a lot was Gothic Revival. This was used for churches, universities, cemeteries, and even residential. The technological advances of the 19th century were shown through the architecture. The mid to late 19th-century became notable for the introduction of skyscrapers. The first "skyscraper was built in New York. The first skyscraper made out of steel was built in Chicago. This paved the way for taller and taller buildings to be made. 

Below is a 21st Century building. It was made in Rome, Italy designed by Richard Meier. 

http://www.archdaily.com/20105/church-of-2000-richard-meier