nginx hands down. nginx is probably the most stable piece of software with the least problems we run here at engineyard. We've had thousands of nginx instances in production for years now and I can truly say it is one of my favorite pieces of software.
I second this recommendation, except on a vastly smaller scale. My uptime currently reads about 530 days. My wee little nginx has not needed a restart in the entire time, except for reloading configuration files. This is through, and this is a back of the envelope number, probably 20 million HTTP requests.
It is far and away the most stable software I've ever seen (at least, among non-trivial publicly-accessible pieces of software).
It also has memory requirements which are low enough that some Japanese toilet seats probably qualify.
