David Akers based his analysis of gold coins on the number of times they appeared at auction and they grades in which they were listed. Let's compare his findings with what we know from the current PCGS Population Report listings.
It turns out the 1857 is considerably rarer overall than the 1854-O, and the 1855 turned out to be twice as common as the 1857. Akers' was spot on in comparing the overall rarity of the 1857 to the 1856, and he correctly noted that both the 1854-O and the 1856 were much more difficult to locate in Mint State. The date that "behaves" the closest to the 1857 is actually the 1859 (which has a much lower mintage).
The 1857 $3 appears most frequently in MS63. MS64 examples are rare and anything MS65 or better is extremely rare. The finest example is a PCGS MS66 in the Simpson Collection.