The 1872-S Quarter Dollar is a relatively common coin, though it is difficult to locate examples in Mint State. The number of certified examples in all grades is smaller than that of scarcer dates (such as the 1872-CC for example) but this may be because lower grade coins are simply not submitted for grading. Thus, users must be careful to consider certified populations in context and not by themselves, as there may be other factors that affect the number of coins that are submitted and graded. Having said that, it is probably safe to assume that the number of certified Mint State 1872-S Quarters is reflective of the overall population because certification does add value in the higher grades.
PCGS has certified three Mint State 1872-S Quarter Dollars, the finest of which is a single PCGS MS66 (finest by a full two points).