What is the correct use of a non-sentence? What is an example?
A full or correct sentence must include a subject and a predicate. The one apparent exception to this is a sentence in which the verb is in the imperative mood, which is indicated in English by the omission of the subject "you". Thus "Please be quiet!" or "Hurry up!" are both complete sentences with verbs in the imperative mood and an unstated subject ("you").
Incomplete sentences are not used in formal non-fiction prose, but are often used in ordinary conversation, text messaging, and in reported speech in fiction or non-fiction. In text messaging, many parts of a sentence may be omitted or abbreviated for brevity. Since, as a genre, it uses a limited number of conventional phrases, readers will normally understand the abbreviations used by their specific peer groups. "lol" as an abbreviation for "[I am] laughing out loud" is generally understood and accepted within the context of internet chat or text messaging but unacceptable in formal writing. The one exception to that would be if embedded with a direct or indirect quotation, for example:
She replied to his long, convoluted message with a single, three letter response: "LOL".
In addition to frequent use in dialogue, non-sentences, or sentences which lack all the appropriate parts of speech to be a complete sentence, can be used for emphasis. Consider the following: Jamie knew that she needed a miracle. A big one. The first sentence is a regular, grammatical sentence. However, the second one is not, but it does emphasize what Jamie needs by simply stating only the most important words in the next sentence. Consider the change if the sentences read like this instead: Jamie knew that she needed a miracle. She needed a big one. It's a lot less emphatic that way because the sentence includes words that aren't as important. If the writer uses the non-sentence instead, the emphasis is much more pointed. Therefore, writers (or speakers) might use non-sentences in order to emphasize the most important idea in a phrase and to avoid diluting it with a bunch of other words that aren't really needed.
If you remember what a sentence is (a subject and a predicate which make sense) and its purpose, a non-sentence can be used in much the same way without all the requirements. A non-sentence is often used in dialogue because people don't always speak in complete sentences. "You must leave now before the soldiers come" is a complete sentence but doesn't convey the emotion attached. A non-sentence such as "Run! Soldiers coming!" conveys the emotion attached, the fear of the speaker, and is more like real dialogue. "Run" can be a sentence with an understood subject of you, but "soldiers coming" cannot be a sentence unless words are added. Think carefully about your use of non-sentences because they can be very useful, but overuse makes for a choppy writing sample.