The public server with most forks hardfought.org. Info on how to connect is on https://www.hardfought.org/nethack/
You can watch in your browser: https://www.hardfought.org/nethack/hterm/
There are 3 servers in Europe, the US, and Australia.
Most people are in various channels (e.g. #nethack, #hardfought) on Freenode but the #hardfought channel is also bridged to a channel on the Roguelike discord server (https://discord.gg/xUPxRe).