Nginx

"nginx [engine x] is an HTTP and reverse proxy server, a mail proxy server,
and a generic TCP/UDP proxy server, originally written by Igor Sysoev."

Website: http://nginx.org/

Build and run instuctions

$ docker build --no-cache -t konstruktoid/nginx:latest -f Dockerfile .
$ docker run --cap-drop=all --cap-add={chown,dac_override,net_bind_service,setgid,setuid} --name nginx -d -P konstruktoid/nginx
c7cd68ec16bf3b480591f441dedf6612d2fbd69f216d432a01208bfa9dacc103
$ docker inspect --format='{{.Config.Healthcheck}}' konstruktoid/nginx
{[CMD-SHELL curl -f 127.0.0.1 || exit 1] 5m0s 3s 0s 0}

There's also a concept version with a static NGINX server running on a limited Busybox in the busybox branch.