Around 60% of all web servers ran Linux, but we don't have any data that could tell which among the Linux distributions are widely used or preferred. If you ask me, here are some of the best and perhaps popular distro for web servers:
Slackware is the oldest surviving Linux distribution so there are no doubts about its reliability. It aspires for design stability and simplicity, using plain text files for configuration and making as few modifications to software packages as possible from upstream. As they say, there is no better, more customizable, standard distro than Slackware.
Gentoo is a highly flexible and fast distro that is built on top of the Linux kernel and based on the Portage package management system. It describes itself as a metadistribution, "because of its near-unlimited adaptability". Unlike a standard software distribution, the user compiles the source code locally according to their chosen configuration in Gentoo. Its package management is designed to be versatile, , portable, easy to maintain, and optimized for the user's machine.
Debian is a strict advocate of the free and open source philosophies. It is known for amplitude of options like its support for a wide range of computer architectures that ranges from the Intel/AMD 32-bit/64-bit to the ARM architecture. Some of the others notable features of Debian are the APT package management system, repositories with large numbers of packages, and the high quality of releases.
RHEL (Red Hat Enterprise Linux)
When we talk about Linux web servers, we shouldn't fail to include RHEL. It is released in server versions for x86, x86-64, Itanium, PowerPC and IBM System z. Though it's proprietary, it costs less than Microsoft's web server software. RHEL is also known for its excellent technical support and service.
If you want to get all the goodness of Red Hat Enterprise Linux but don't want to spend a dime, then CentOS is for you. CentOS is based on RHEL and aims to provide a free enterprise class computing platform and strives to maintain 100% binary compatibility with its upstream distribution. Technical support is mainly provided by the community through its official mailing lists, web forums, and chat rooms.
But don't just limit your choices from my list, as there are other distros that are as capable as the above mentioned.