checkbashisms-compliant way to determine the current shell

Usually one uses $0 for this purpose. On the site which you linked it stands:

0
   (Zero.) Expands to the name of the shell or shell script.

Your

# If the current (login) shell is Bash, then
if [ "${BASH_VERSION:-}" ]; then
  # source ~/.bashrc if it exists.
  if [ -f "$HOME/.bashrc" ]; then
    . "$HOME/.bashrc"
  fi
fi

code is completely POSIX conformant and the best way to check that you're currently running bash. Of course the $BASH_VERSION variable is bash-specific, but that's specifically why you're using it! To check that you're running bash!

Note that $BASH_VERSION will be set whether bash is invoked as bash or sh. Once you've asserted that you're running bash, you can use [ -o posix ] as an indicator that the shell was invoked as sh (though that option is also set when POSIXLY_CORRECT is in the environment or bash is called with -o posix, or with SHELLOPTS=posix in the environment. But in all those cases, bash will behave as if called as sh).

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Another variable you could use instead of $BASH_VERSION and that checkbashism doesn't seem to complain about unless passed the -x option is $BASH. That is also specific to bash so one you should also be able to use to determine whether you're running bash or not.

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I'd also argue it's not really a proper use of checkbashisms. checkbashisms is a tool to help you write portable sh scripts (as per the sh specification in the Debian policy, a superset of POSIX), it helps identify non-standard syntax introduced by people writing scripts on systems where sh is a symlink to bash.

A .profile is interpreted by different shells, many of which are not POSIX compliant. Generally, you don't use sh as your login shell, but shells like zsh, fish or bash with more advanced interactive features.

bash and zsh, when not called as sh and when their respective profile session file (.bash_profile, .zprofile) are not POSIX conformant (especially zsh) but still read .profile.

So it's not POSIX syntax you want for .profile but a syntax that is compatible with POSIX (for sh), bash and zsh if you're ever to use those shells (possibly even Bourne as the Bourne shell also reads .profile but is not commonly found on Linux-based systems).

checkbashisms would definitely help you find out bashisms but may not point out POSIX syntax that is not compatible with zsh or bash.

Here, if you want to use bash-specific code (like the work around of that bash bug whereby it doesn't read ~/.bashrc in interactive login shells), a better approach would be to have ~/.bash_profile do that (before or after sourcing ~/.profile where you put your common session initialisations).

Usually the SHELL environment variable tells you the default shell. You should not need to manually source the .bashrc file (not true see update bellow), bash should do this automatically just make sure it is in the $HOME directory.

The other option is to do something like:

 ps -o cmd= $$

that will tell you the command (without arguments, the = without a value will not display column headers) of the current process. Example output:

 $ps -o cmd= $$
 bash
 $sh
 $ps -o cmd= $$
 sh

UPDATE:

I stand corrected! :)

The .bashrc is not always sourced as was mentioned in the comments bellow and https://stackoverflow.com/questions/415403/whats-the-difference-between-bashrc-bash-profile-and-environment

So you could move your .bashrc to .bash_profile and see if that works without having to do the test. If not you have the test above.