I'm an undergraduate student (United States) majoring in X, applying to PhD programs with a research focus in subfields Y and Z.
I conjecture that the best universities for Y and the best universities for Z are not necessarily the same, and I want to narrow down my PhD applications to universities that are known for one or both of these subfields.
None of the professors at my university's X department do much research in either Y or Z, and they seem to not have the greatest idea where I should apply - I usually get generic answers like Prestigious University U, which may or may not be what I'm looking for.
The other concern is that I've taken a few graduate classes regarding Y and Z, but in no way am I fluent in the fields, nor have I done direct research in these fields. As such, it would be difficult for me to attempt to read research papers about Y and Z to try to back out which universities produce the best work.
How might I go about finding the strongest programs in Y and Z?
Look at the research publications of faculty in your area of interest and then look for the most widely cited research in this area. You can easily find such information using tools like Google Scholar. Then find out where the authors of these widely cited papers are teaching.
Look at what happens to graduates from the different programs. Many faculty have web pages that list their recent PhD students, some universities publish graduation programs that include lists of graduates, and there are sometimes other ways of identifying recent graduates of a particular Phd program. If you can't find those people on the internet, that's a strong indication that they working in academic careers. If you find them but they're teaching at community colleges and regional comprehensive universities, then that tells you a lot about how well the graduates have done in academic careers.