I am wondering how many research interest should be included in a statement of purpose. Is it better to include just one core interest or to include up to three as in my case?
Be as specific as possible. Do not bluff.
Remember that admissions committees are looking for strong evidence of research potential. One of the markers of that potential is a deep interest in your intended research area. For that reason, it's important to describe your potential research interests in specific and credible detail. Why are you interested in field X? What specific problems are you interested in working on? What projects have you done? What papers have you read (or written)?
It doesn't matter all that much what you write about. We know that your interests will change over time. Nobody is going to limit you to the specific research topics you describe in your statement. Your statement is at least as much a demonstration of intellectual maturity as it is a description of research interests.
Eykanal's observations are correct. Most graduate school applicants "barely know what's being researched". A list of buzzwords mined from faculty web pages is not credible. You can't effectively describe what your interests are when you aren't familiar with the field. But I disagree with his conclusion; just because it's hard doesn't mean you shouldn't do it. Don't be most applicants. Know what's being researched. Don't be vague, and don't just list buzzwords. Make yourself familiar with the field.
After you've done that, writing about your research interests is easy, because you actually have some.
One big problem with writing research interests for graduate school is that you barely know what's being researched. How could you state definitively what your interests are when you aren't familiar with the field?
With that in mind, if you're a member of the minority of undergrads who actually performed undergraduate research and knows what you want to research, feel free to list a single field of interest. If you can list specifics (e.g., "Following up on both the research I did in last summer, as well as the research from the Smith lab in Princeton, I would like to examine...") that would be even better, as it would demonstrate that you know what you're talking about. (Pro tip: make sure you know what you're talking about when writing it.)
If you're like the rest, though, you don't know the field, and that's OK. It would probably be better to give a really vague statement than to list a few things. Compare the following:
I'm interested in furthering my knowledge of the biomedical engineering field.
I'm interested in the fields of brain-computer interfaces, tissue engineering, and medical device design.
This is subjective, but to me both statements say the same thing, with the first one being concise and the second one sounding like buzzword soup.