UPDATE 2018: Cloudflare announced 18.104.22.168 public DNS Service.
I haven't been using the DNS that my ISP provides for five years now and many problems have gone away. Especially problems regarding updating speed. I've been using OpenDNS for my systems, and Google DNS for most of the setups I do for other people. Both because they are stable and easy to remember.
Lately I decided to find out what other option are out there. I'm happy to provide a complete list of IPv4 and IPv6 of publicly available DNS.
# OpenDNS Family Shield
# Google IPv4 nameservers
# Basic Yandex.DNS
# OpenDNS IPv4 nameservers
# Comodo nameservers
# Cloudflare IPv6 nameservers
# OpenDNS IPv6 nameservers
# Google IPv6 nameservers
All of the above are free to use at the time of this writing. You can either configure them on your computer or on your router. I'll let you figure how to do it yourselves ;)