There are actually many fine uses for sentence fragments. Here’s a brief scene from an imaginary Greek tragedy composed entirely of fragments:
Menelaus: Aha! Helen!
Helen (startled): Beloved husband!
Paris (entering, seeing Menelaus): Oops. ’Bye.
Menelaus: Not so fast! (stabs Paris).
Some people get into trouble by breaking a perfectly good sentence in two: “We did some research in newspapers. Like the National Enquirer.” The second phrase belongs in the same sentence with the first, not dangling off on its own.
A more common kind of troublesome fragment is a would-be sentence introduced by a word or phrase that suggests it’s part of some other sentence: “By picking up the garbage the fraternity had strewn around the street the weekend before got the group a favorable story in the paper.” Just lop off “by” to convert this into a proper complete sentence.