The 1851 Half Cent has the highest mintage of the Braided Hair type but it is not the most commonly encountered date (that distinction goes to the 1853). In Mint State, the 1851 Half Cent is fairly common, but not as common as dates such as 1853, 1854, and 1855. The finest certified example is a single PCGS MS66 Brown. Most Mint State examples seen are in the MS-63 to MS-64 range. In full Red, more of the 1851 Half Cent have been certified than any other.
On high grade examples, the bottom of a misplaced 1 can be seen to the right of the date.
Only one die variety (Cohen 1) is known of the 1851 Half Cent.
Sources and/or recommended reading:
"American Half Cents, Second Edition" by Rogern S. Cohen, Jr.