Are citations in abstracts considered bad style?

It is not entirely unreasonable to include a citation in an abstract, if the reason you are citing it is because your paper is a major extension, rebuttal, or counterpoint to the cited article.

In that case, however, you do have the responsibility of providing the reference within the body of the abstract. For example,

We extend upon the results of Smith [Journal of Very Important Results, 1, 374 (2012)] to include the effects of a doohickey at the end of the thingamajig.

In such a case, the abstract remains self-contained, with an important citation included. (This is especially essential if an author is well-known for multiple papers, in which case the reference can be used to distinguish the varous works that could be intended.)

Large numbers of citations, however, should be avoided, as should "secondary" citations. Only the most critical literature for a paper should be cited, and that should normally be limited to one or two. Any more than that, and the abstract becomes hard to read.

This issue may depend on the academic field of the publication. In the case of the social sciences, abstract are generally written to be independent of the other sections of the paper or manuscript, so citations in the abstract are avoided. You may include a citation, but sometimes you have to include all the bibliographic details. Considering that abstract are usually required to be short, you may be unnecessarily wasting words.

Moreover, as the abstract is intended to be an interesting summary of the research described in the manuscript, it is not probably useful to include citations. An exception is the case when a manuscript heavily draws on a previous work. For instance, if you are replicating a previous study, then you may have to include a cite. In this paper, the author replicates and extends a study. The title and the abstract have a citation of the previous study.

It depends on the situation.

An abstract for a paper must be stand-alone, because the bibliography is hidden in the paper itself. The abstract must contain all information required for people to judge if they want to read the paper, and as there is no bibliography, the reader does not know what the citations relate to. Therefore, there should be no citations.

It may be a bit different if the abstract is for a conference. Maybe in some situations it's possible to add one or two references at the end of the abstract. In that case, it can be okay to have citations.

So the most important question here is: can the reader use the abstract as a stand-alone unit? If using citations cause the answer to this question to be no, don't do citations.