From what I've found, there isn't really any solid research supporting the existence of learning styles, but there is solid research refuting them. It is true that plenty of people self-report different learning styles, but nothing has shown that their beliefs have a basis in fact or biology.

This article nicely summarizes the evidence that learning styles don't exist. Here's a quote:

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My job was to run classrooms full of Navy enlisted men through aircraft identification training. Enlistees were randomly assigned to classes that emphasized one style of training or another. However, all classes received equivalent amounts of practice and feedback. No significant differences in acquisition or retention were discovered across groups regardless of enlistees' preferences for particular learning styles. This finding, resulting from methodologically rigorous research—in which I played a long, boring supporting role—vividly illustrated to me the trouble with learning styles: They don't work.

Another quote from the same article:

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Unless I, and the other reviewers of research in this area, have missed the publication of tons of replicated findings, there is no evidence of stable interactions between learning styles and instructional methods. Why then do otherwise knowledgeable educators and educational researchers persist in making unverified claims for learning styles? I can only conclude that they adhere to what Jeanne Chall (2000) in her last book called a romantic, as opposed to rational, view of education. Chall cites other romantic notions that have little verified empirical support such as the whole-language approach to reading instruction, open education, and discovery learning, to name only a few. Sometimes an idea may appear so logical, and/or so deeply related to the values held by individuals, that it becomes an article of faith. Believers cling to their fancies irrespective of research findings.

There's also a field of research called denial science. It investigates why people believe stuff in the face of evidence to the contrary. It's very interesting. Here's a summary of current ideas.