How can you tell when a sentence is a sentence fragment?

A sentence with no subject or verb is an incomplete sentence. That would be a sentence fragment.

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A complete sentence requires a subject and that subject'spredicate, excluding any subject/predicate pairs that are part of a dependent clause.

Sentence fragments often appear as short answers to questions, where the subject or predicate is already known. This is acceptable in speech but considered incorrect writing.

Ex Q: "What are you doing?"

Ex A: "Running home."

The answer is a sentence fragment because it has no subject ("home" is an object; it's not what's doing the running).

Ex Q: "What are your hobbies?"

Ex A: "Hiking, boating, and watching people climb trees."

The answer is a sentence fragment because it has no subject. Notice that it contains a complete subject-predicate pair ("people climb"), but that pair is part of an object clause.

Ex Q: "Who keeps poking me in the back?"

Ex A: "Stefan, with that pencil he just sharpened."

The answer is a sentence fragment because it has no predicate. Again, there is a subject-predicate pair ("he sharpened") but that pair is part of a dependent clause.

Special case: Imperative

Many languages allow the omission of a subject because the verb conjugation implies only one (or very few) matching pronouns. In English, that's not usually the case, except with commands. When you write a command, the subject "you" can be left out:

Ex: "Stop goofing off!"

The implied subject is 'you' and the predicate is 'stop'. This is a valid sentence.