These are dependent clauses that can't normally stand on their own.
During the show's premiere is a fragment. Who's doing what during the show's premiere? You can complete the thought by joining it to a sentence containing a subject: During the show's premiere, the man in the white coat stood up to play his kazoo.
Fragments can, however, be used for emphasis: He wouldn't fix his house. Not yet. Not until after the storm. But use them judiciously; too many fragments can make your writing feel stilted and look unprofessional.
Sentences. That are not complete. But punctuated. As if they are. Are called sentence fragments.
Plainly put, a sentence fragment is part of a sentence that usually cannot stand on its own. Either it is a dependent clause, or it's not a clause at all. A clause, which must have a subject and a predicate, is dependent when it has words like "when" or "if" in front. Dependent clauses have to lean on another clause. In other words, a sentence fragment will either lack a noun, noun phrase, or a pronoun that denotes the doer of the action, or what's being said about the subject, which includes the verb and any modifiers that are needed.
Consider this complex sentence: During the show, Frankie sat up and began to yodel. Which part would become a sentence fragment if left to stand on its own? During the show, of course. It's incomplete; it lacks a certain element needed to be a complete sentence: the predicate. The other, Frankie sat up and began to yodel, is complete. So, written in a piece of prose as “During the show. Frankie sat up and began to yodel.”, one of them would be considered grammatically incorrect.
But sentence fragments are sometimes used for emphasis. Take our prior complex sentence, for example. Placed after the independent clause, the sentence fragment once considered grammatically incorrect would now be considered acceptable: Frankie sat up and began to yodel. During the show! The fragment is telling the reader where and when Frankie began to yodel, with emphasis on how inappropriate his yodeling was.
Yes, there are going to be times when a writer needs to write in sentence fragments. People think in fragments. People speak in fragments. With some exception, a formal straight narrative usually calls for complete sentences to convey a meaning or an idea, but during interior monologue and especially in speech, sentence fragments are allowed. So an After a while, he thought with with a shrug, or a “Hey you, with the stupid-looking face!” or even a simple, exclamatory, “No!” are all acceptable sentence fragments.
But woe to the writer who doesn't understand the difference between a sentence fragment used properly and one that is wandering alone, confused and misplaced, in one's work. Abused, sentence fragments can drown an otherwise coherent piece beneath a series of awkward, stilted sentences. Used to their best advantage, they can help a writer looking to vary the cadence of his piece, add emphasis where needed, or get a point across quickly.
So the sentence fragment need not be shunned, but rather learned about and embraced as a useful tool. Thus, such is the life of a sentence fragment. Simple. Short. To the point. The end.