Define modern. If it means 20th Century: Standing Liberty Quarter If it means post 1964: Sacajawea Dollar (reverse)
The Walker design followed by the Mercury dime and the Standing Liberty quarter. All started in 1916 and used over really crazy times in US history.
My favorite designs are the art deco coins from 1907 and 1916-1921. The draped bust dollars of both eagle types are a very close second. The 1793 Wreath cent, 1793/4 Liberty Cap Cents, and capped bust gold are third.