Your teachers hated them. Word processors despise them. And we writers love them. But are they useful?
First we ignore the rule - yes, that rule we learned back in school during English class. Every sentence must have a subject and a verb. It's a nice rule but does it always apply?
He looked up in the sky. What is it? A bird? A plane? No, it's Superman! Come to save the day!
Quick fragments blasting from the past to show us the way. "A bird? A plane?" and a longer last one. All are fragments and should have been expanded to "Is it a bird? Is it a plane? and He has come to save the day!" Now the sentence is no longer a fragment, it has a subject and a verb. All the English teachers are happy and content.
But, let's take a repeat of the corrected error...
He looked up in the sky. What is it? Is it a bird? Is it a plane? No, it's Superman! He has come to save the day!
Somehow, by making the sentence grammatically correct, it has lost its punch, its vitality. The impact of the moments is missing. Anticipation is missing.
This holds true for your writing. When composing a scene with action, use short, fragmented sentences to create and impart a sense of urgency. Long, drawn out descriptions and correct grammar slow down the reading.
Azbar's gold handled sword with the ruby encrusted hilt flashed through the air. It slashed the silken coverings of the pillow. His hand moved left, the sword's edge catching a glint of the lantern's light. The enemy twisted the green dagger in the low light, feinting right before lunging.
A very simple scene. Now, a little rewrite.
Azbar's sword flashed through the air. Feathers flew from a pillow. Left. Light glinted on the sword. A dagger twisted. The enemy feinted right before lunging.
Not the best examples, but... Basically the same scene. The 2nd attempt gave more imagery by making you see the movement quick and fast. I bet the feathers are still floating about in your mind which filled in the blanks. Do you really need to know the sword's handle was gold and the hilt was encrusted with rubies? Silk pillow? Cotton pillow? Does it matter if the dagger is green or maybe blue? Bottom line, a dagger twisted.
Fragments. Sentence mini-clusters can give more information than a long line of skillfully smithy-ed words. These fragments allow you, the writer, to convey images quickly to the reader.
Do you want -- He jumped into the pristine, cold, fresh waters of the deep, azure-blue lake.
Or do you want -- He jumped into the lake. Cold. Fresh. Pristine. Deep.
Remember, fragments can be your friends, especially if used properly and only when necessary.