I like the Draped Bust design of 1800-1808 the best. It's a copy of the 1796-1807 Large Cent design, but I think it's much prettier in its diminutive form.
Image Courtesy of: Heritage Auction Galleries, Ha.com
The smaller size makes Liberty seem sweeter and the reverse wreath is more delicate, at least to my eyes.
Before 1803, many of these were struck on cut down planchets of rejected large cents and sometimes the large cent design is actually visible. Many will show die misalignment like my 1806:
That's so common in these early coins that it carries no extra value and often won't even be mentioned unless really pronounced.
The true rarity of these coins is evidenced by the fact that the 1806 pictured above is considered rather common for the series, yet the original mintage of all varieties of 1806 was only 356,000 and it is estimated that only a few thousand survive today. A few hoards of draped bust half cents have been discovered, but only of a few hundred pieces, so they barely deserve the name.
If you are interested in these under appreciated coins, I recommend getting a copy of "Walter Breen's Encyclopedia of United States Half Cents, 1793-1857". It's out of print, but copies do turn up on Amazon.
Note: All my coins are in a safe deposit box. I keep nothing in my home.
Contests and giveaways: http://coins.aplawrence.com/2014/05/contests-and-giveaways.html