What is the most secure hand cipher? By "hand cipher", I mean a symmetric cipher for which encryption and decryption can can both be performed with a pencil on graph paper, consuming about 10-20 seconds per character by a proficient user. Additional simple tools, e.g. a deck of cards or a multiplication table, may be used to expedite the process. No mathematical proficiency is required other than mental addition, subtraction, and maybe multiplication.

Basically, it should be a cipher that someone of average mathematical abilities can master, and it should require no digital technology or cryptographic tools that cannot be constructed out of paper.

For example, four-square in CBC mode seems secure to me because CBC is a well established mode of operation and four-square seems to be a good block cipher as it accommodates large keys and seems to produce a random-looking output. However, the fact that four-square in CBC was never widely adopted before digital encryption became a consumer technology leads me to believe that someone broke the cipher and thus discouraged people from using it.

So is there any hand cipher that the best supercomputers would take years to cryptanalyze? If not, then for which hand cipher is the best known attack the most complex?
Solitaire by Bruce schneier is probably your best bet.  It has a few issues but it will work well for most things.  It ends up having a small bias, but it takes about 15 seconds per character after the initial key stream has been generated.

It is not nearly as widely studied a field as most people are assumed to be able to get a computer or other computation device.  The only other one I can think of is VIC.  It is based on a lagged Fibonacci generator which is a bit of a different place to start from. See this other answer for more details on that cipher.

I'd strongly suggest looking long and hard at Solitaire. It has really a cool design.