"1993 Feb. 26, New York City: bomb exploded in basement garage of World Trade Center, killing 6 and injuring at least 1,040 others. In 1995, militant Islamist Sheik Omar Abdel Rahman and 9 others were convicted of conspiracy charges, and in 1998, Ramzi Yousef, believed to have been the mastermind, was convicted of the bombing. Al-Qaeda involvement is suspected.

And:
1995 April 19, Oklahoma City: car bomb exploded outside federal office building, collapsing wall and floors. 168 people were killed, including 19 children and 1 person who died in rescue effort. Over 220 buildings sustained damage. Timothy McVeigh and Terry Nichols later convicted in the antigovernment plot to avenge the Branch Davidian standoff in Waco, Tex., exactly two years earlier. "
http://www.infoplease.com/ipa/A0001454.html
These terrorist attacks occurred on America soil, though I don't know why you should limit it to America soil.