I've heard that in some states you can actually apply to become a resident after completing your first year in dental school. Some states indicate that you cannot become a resident for tuition purposes if you came to that state for education purposes. Each state is different. If you find a list of which is which, post it on sdn.
Of course, what is ideal is to be a resident at the time you apply to a state school. Most states have requirements that all/most of their accepted students be residents of that state at the time of application. The remaining slots are open to all other applicants-some schools do not save any slots for non-residents. Competition is therefore extremely fierce for the non-resident slots. (To be fair, competition is fierce even if you are a resident but the odds are very stongly against the non-resident) Some state schools also consider your connection to that university outside of residency such as where your relatives went to school and whether any of your family contributes to the alumni fund. If you make it to the secondary application, I suggest you spell your connection out as clearly as possible. Everyone wants you to get in but your sister-in-law, the chair of board of regents, thinks you'll make a great dentist.
So what it boils down to is that approx 14,000 students are applying for dental school to approx 50 dental schools in the US. Some schools have small admitted class size and some have large admitted class size. I think NYU's admitted class size is around 300 and is toward the high end. Some states do not have a state dental school. I suspect that the vast percentage of applicants are applying for the small percentage of slots at state schools open to non-residents and slots at private dental schools. I've heard that the average number of schools an applicant applies to is in the neighborhood of 9.