How to change default shell to ZSH - chsh says "invalid shell"

Add zsh to /etc/shells:

command -v zsh | sudo tee -a /etc/shells

You can now use chsh to set zsh as shell:

sudo chsh -s "$(command -v zsh)" "${USER}"

See this documentation: Changing your login shell

You need to add /usr/local/bin/zsh to /etc/shells.

First, check if zsh is listed as a valid shell by

cat /etc/shells

If zsh is not listed, install it. For example, if you use apt

sudo apt-get install zsh

Do, step 1 again and see the path of zsh. In my case, both /usr/bin/zsh and /usr/zsh is listed. As @Stéphane Chazelas mentioned in the comment, if you like to compile and install the shell from scratch, be sure to add the path in /etc/shells.

Change the shell using

chsh -s /usr/bin/zsh

From man page:

-s, --shell SHELL The name of the user's new login shell. Setting this field to blank causes the system to select the default login shell.

Logout and login again.