This is a bit tough. First, there are tons of internships, and there is no ranking system. It's a bit hard to evaluate them based on the number of applicants received, as T4C explained, because the number of applicants is heavily influenced by location. Second, different types of internships are better suited for different people. Generally, the more strongly research oriented ones (e.g., Brown, UIC, Western Psych) are considered more competitive, but on the other hand, someone who is a good match for one of those internship (through publications and grants) may not have a chance at a more clinically oriented internship (because of insufficient hours). So your top internship could differ from another person's. Finally, a lot of people go to internships based on location rather than quality. It is a pain to move across the country for a one year gig when you are 29, married, and have an infant, and/or own a house (as an example). So, a lot of people sacrifice internship quality because of life demands. And truly, it doesn't matter that much. Internship doesn't impact your career nearly as much as other factors.
The closest thing to a ranking system that I know of (and if anyone knows of a better one, I'd love to be pointed toward it-- I'm getting to the stage where I'm thinking about where to apply fro internship) is this website:
http://psych.arizona.edu/apcs/members.php
It's the website of the Academy of Psychological Clinical Science. The member programs that it lists are all high-quality, research-oriented programs, which would be considered competitive (it also lists grad programs). There are, however, plenty of other good internship sites in addition to these.
Boston Consortium in Clinical Psychology
Brown University Medical School Consortium
University of Illinois at Chicago, Department of Psychiatry
Palo Alto VA Center
Medical University of South Carolina
VA Maryland Health Care System /University of Maryland Internship Consortium
University of Washington School of Medicine
Western Psychiatric Institute and Clinic
University of Wisconsin School of Medicine