In an ideal world, scientific papers wouldn’t have abstracts.
There’d be no need for them, because every ideal scientist would have the time to read every published paper in full, and a perfect memory for all of the details.
Sadly we don’t inhabit such a world, so abstracts are perhaps the most important part of a manuscript – certainly so, in proportion to the word count.
The bottom line is, if you as an author want citations for your paper, you need a good abstract. Yet abstracts often go wrong. In my opinion (and I read a lot of the things), an abstract should not contain…
Prologue. Many good abstracts begin in media res, describing the methods with no introduction at all. Alternatively, a concise intro can work, but this should be limited to explaining the question that the study set out to answer. A list of prior findings is not called for. A paper should start with a review of previous literature – but an abstract is not a paper, nor is it the beginning of one. It’s a whole art form in itself.
Footnotes. A paper’s abstract [1,2] just doesn’t need  these . I think they generally end up there because of hasty copy-and-pasting from conference abstracts, where they’re common. Anyway, avoid doing this. But if you insist on having them, at least make sure the notes they refer to are included at the end of the abstract!
Promises: An abstract should make the reader want to see the full paper. It shouldn’t tell them to do so – rather an abstract should show readers how interesting the paper is, by being interesting itself. So avoid empty phrases like “Implications will be discussed…” or “Several results were observed…”. If a result is interesting, describe it; if not, don’t mention it.
Mistakes. You might get away with a few English language errors or typos in the main text, but you just can’t afford even one in the abstract. Anyone who notices it won’t bother to read (or cite) you; the rest of the paper could be perfect, but that’s no good, if no-one reads it. Oh, and make sure the statements in the abstract match the ones in the paper.