If we're talking specifically about executable scripts, here is #bash's (libera.chat) factoid on the matter:

Don't use extensions for your scripts. Scripts define new commands that you can run, and commands are generally not given extensions. Do you run ls.elf? Also: bash scripts are not sh scripts (so don't use .sh) and the extension will only cause dependencies headaches if the script gets rewritten in another language. See http://www.talisman.org/~erlkonig/documents/commandname-extensions-considered-harmful

It's for these reasons that I keep my executable scripts named without extensions (e.g. install).

I sometimes have non-executable scripts: they're chmod -x, they don't have a shebang, and they're explicitly made for source-ing (e.g. library functions). For these, I give them an extension depending on what shell I wrote them for (and thus, what shell you need to use to source them), e.g. library.bash or library.zsh.

Just put the shebang at the top of your script:

#! /usr/bin/env bash

I'm not a big fan of extensions because if you put the script in your $PATH it's weird to type do_the_thing.bash