Sentence fragments are often considered mistakes when it comes to writing. However, there are instances in which sentence fragments can be used to good effect. The video below describes sentence fragments and offers a couple of guidelines on when their use is acceptable.
Source: Dan Reade
When it comes to writing, in general it is best to avoid sentence fragments. Most readers find them distracting. Most instructors will mark them as wrong. However, like anything in writing, there are times when it makes sense to break the rules. Here are four potential instances in which a writer may choose to use sentence fragments:
"I can't believe you would do that. My own brother!"
"My own brother" is a sentence fragment. However, in this instance it's used to accentuate the speaker's disbelief.
"Is Tom Brady that best quarterback to ever play football? Not according to Sid Smith."
"Not according to Sid Smith" is a sentence fragment, but because it answers the previous question, it fits smoothly within the paragraph.
"For this reason, Johnson states that drinking is wrong. Now for the other side."
"Now for the other side" is a fragment, but it helps show the reader that the writer will be transitioning to a different argument.
"No!" Then he wept.
"No" on its own is a sentence fragment, but because it's being used as an exclamation, it works in this instance.
While sentence fragments can be used in these situations, it's still a good idea to use them sparingly. Also, make sure any use is intentiional; the unintended sentence fragment can really confuse readers. Last, for graded assignments, always check with an instructor first. Writers and readers disagree about many things, including when sentence fragments are acceptable, so if a paper is being turned in for a grade, it's always best to see if an instructor agrees with these guidelines.
Source: Hacker, Diana. "Rules for Writers." 6th ed. Boston: Bedford/St. Martins, 2009.; Dan Reade