In 1797, the U.S. Mint made Half Dimes with differing numbers of stars on the obverse. Some were made with thirteen stars, some with fifteen, and others with sixteen. The reason for this numerical difference was because of the original practice to include a star for each state admitted to the Union. After the 13 original states came Vermont, then Kentucky in 1792, and Tennessee in 1796. Clearly, there were timing issues. For instance if a 1797 die were made in 1796 in anticipation of the next year's production, it may have had either 15 or 16 stars, depending on whether it was made before or after Tennessee was admitted. However, either way, Mint employees soon learned that the stars were becoming awfully crowded on the coins, particularly on the small Half Dimes, and they eventually reverted back to 13 stars -- and kept it that way (with a few minor exceptions).
The 15 Stars variety is the most common of the three major varieties of 1797, by a large margin. Though the exact mintage is unknown, the PCGS Population Report gives a good estimate of relative rarity. Currently, more than two times as many 15 Stars 1797 Half Dimes have been graded than the other two major varieties combined. The majority of certified examples appear in circulated grades. In Mint State, PCGS has certified over two dozen examples, the finest of which include one MS66 and one MS67.
In general, the obverse of the 1797 15 Stars Half Dime shows a good strike, though sometimes the central details can be weak. Usually, the reverse is poorly detailed in the centers, this being the high point of the reverse. Adjustment marks are seen far less frequently on the Half Dimes than they are on the larger denominations, so this is seldom an issue on this date.