Mark Watkins now reports that his computer has proved 1 e3 c5 to be a win for White, leaving only 1...b6 as a possible non-losing reply to 1 e3. I quote, slightly edited, from an e-mail he recently circulated.
"I write to say that 1. e3 c5 has been solved as a White win under either FICS or International rules (in fact, I use the "joint" rules, where a game is a draw unless it is a win under both). This leaves b6 as the only possibly non-losing response for Black.
"The page http://magma.maths.usyd.edu.au/~watkins/LOSING_CHESS/e3c5.html has some running commentary, and piecemeal proof files. The main page: http://magma.maths.usyd.edu.au/~watkins/LOSING_CHESS/ has the latest source code download and the full proof file.
"Klaas Steenhuis also recently solved 1. c3 e5 and 1. f3 e6 as Black wins, and these are available from the above LOSING_CHESS webpage."
So only one to go...