I have no problem with sentence fragments. No problem at all. :)
What I have a problem with is fragments that serve no purpose. Here's an example:
The editor was always bitching. About all sorts of things.
Now from a very accomplished writer-- you know, the type of writer whose voice just hums with authority, who can be counted on to experiment with prose and syntax and KNOW when it works and fix it when it doesn't-- I might think, "Oh! This very accomplished author wants me to read that differently than if it were one complete sentence!" And then I'll read it -that- way, and (rather like with enjambment in poetry) get both the broken up meaning and the complete meaning. So I'll read a great deal of exasperation there, even weariness. The editor was ALWAYS bitching. And then, I'll imagine a sigh, and the speaker/character/narrator wearily going on and starting to list all the many things the editor bitched about.
(This "reading"will be more likely if there is a series of fragments, a list of the bitch-topics, btw:
The editor was always bitching. About all sorts of things. About sentence fragments. About "everyone" as a singular noun. About dangling modifiers. Especially about dangling modifiers.)
If the author is trying replicate the staccato quality of some narrator's speech patterns, hey, I can go for that! I'm nothing if not avant-garde. :) But I won't believe that's what is happening if the rest of the manuscript is written in serviceable, not inspired, prose.
There's nothing wrong with serviceable prose, of course. It serves its purpose and doesn't get in the way of the story, but anything outre (there's an accent mark there, but I don't know how to do it in html and I'm too hungry to go look it up!) in serviceable prose is likely to be interpreted as a mistake, not as an experiment. And, well, that's because it usually IS a mistake.
So if you can identify a sentence fragment in your own passage and explain (as I just did about the bitchy editor) how the reader will "hear" it, why it adds to the reader's experience, and why it's worth violating sentence rules, okay! I'll be right there with you, egging you on. "Your narrator is a teenaged girl! So why not put in a 'duh' or two there, just to make the narrative sound more like her?"
(I tell my students that if they use a sentence fragment, they have to put an asterisk there and then at the end of the paper, explain WHY they made that a fragment and why it works better than a complete sentence would. If you think that's evil, keep in mind that when I started teaching 20 years ago, we were told to fail immediately any paper with a sentence fragment. That was unofficial policy. See how much nicer I am now?)
But more often, what I see are sentence fragments that serve no purpose, that the author doesn't recognize as sentence fragments, that add nothing to the experience of reading (except annoyance). Those are the ones I quickly edit, usually by linking them right to the previous sentence, or replacing "which" with "this," or something else the author could just as easily have done himself.
Here are a few examples, culled (and modified) from submissions and student papers.
John was about to leave. Although he wanted to stay.
I already knew the truth. The truth being that she didn't love me anymore.
She pulled off her jacket. Her cardigan too. (This actually might work, if the rest of the narration seemed conversationally adroit. So... why would that work?)
Gas prices were rising through the roof. The stock market falling.
Joella wasn't easily shocked now. By swearing. (See, if the fragment was "By anything," I'd think that it was meant to be emphatic, and thus purposeful.)
Too many people had already emigrated. Because of the potato famine.
She decided to go to dinner. With the group of skateboarders.
He turned the ignition on. Which started right up.
So how about some examples in comments? Most of the above, I'd just edit to connect the sentences. But see, there are two examples that made me speculate. Under what circumstances would you use a sentence fragment, and how would you explain it? Can you give an example? Or would you never use one if you could help it?
(Let me say that I don't consider dialogue as needing a justification for non-standard English, as long as the dialogue works. But even in dialogue, fifteen sentence fragments in a row will sound a bit... jagged. Theresa, please ignore ellipsis. Or if you can't, here's my justification. I wanted it clear that I was hesitating, searching for an adjective, and that perhaps I'd discarded a more nasty one.)
Something to remember: Many of your readers are likely to be sophisticated readers, who read more than just the words to get the meaning. They're reading the rhythm, the structure of sentences, the punctuation, the word choice, the ratio of black ink (or pixels) to white paper (or screen). They're even reading the shape of words. All this can add to the meaning of the passage. They will love it when you exploit their greater meaning-making ability to good purpose. But their greater sensitivity means that they will be put off by signifiers that are NOT to a purpose, because they will react with a meaning-making that turns out to be unfulfilled-- meaning interruptus! (Tell me you know what I mean!)
And how do you know if you are at the level where you can play with those sensitive readers, or if you should stick with serviceable and correct prose?