choosing between $0 and BASH_SOURCE

^{Note: For a POSIX-compliant solution, see this answer.}

${BASH_SOURCE[0]} (or, more simply, $BASH_SOURCE^[1] ) contains the (potentially relative) path of the containing script in all invocation scenarios, notably also when the script is sourced, which is not true for $0.

^{Furthermore, as Charles Duffy points out, $0 can be set to an arbitrary value by the caller.
On the flip side, $BASH_SOURCE can be empty, if no named file is involved; e.g.:
echo 'echo "[$BASH_SOURCE]"' | bash}

The following example illustrates this:

Script foo:

#!/bin/bash
echo "[$0] vs. [${BASH_SOURCE[0]}]"

---

$ bash ./foo
[./foo] vs. [./foo]

$ ./foo
[./foo] vs. [./foo]

$ . ./foo
[bash] vs. [./foo]

$0 is part of the POSIX shell specification, whereas BASH_SOURCE, as the name suggests, is Bash-specific.

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[1] Optional reading: ${BASH_SOURCE[0]} vs. $BASH_SOURCE:

Bash allows you to reference element 0 of an array variable using scalar notation: instead of writing ${arr[0]}, you can write $arr; in other words: if you reference the variable as if it were a scalar, you get the element at index 0.

Using this feature obscures the fact that $arr is an array, which is why popular shell-code linter shellcheck.net issues the following warning (as of this writing):

SC2128: Expanding an array without an index only gives the first element.

On a side note: While this warning is helpful, it could be more precise, because you won’t necessarily get the first element: It is specifically the element at index 0 that is returned, so if the first element has a higher index – which is possible in Bash – you’ll get the empty string; try 'a[1]='hi'; echo "$a"'.
(By contrast, zsh, ever the renegade, indeed does return the first element, irrespective of its index).

You may choose to eschew this feature due to its obscurity, but it works predictably and, pragmatically speaking, you’ll rarely, if ever, need to access indices other than 0 of array variable ${BASH_SOURCE[@]}.

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Optional reading, part 2: Under what conditions does the BASH_SOURCE array variable actually contain multiple elements?:

BASH_SOURCE only has multiple entries if function calls are involved, in which case its elements parallel the FUNCNAME array that contains all function names currently on the call stack.

That is, inside a function, ${FUNCNAME[0]} contains the name of the executing function, and ${BASH_SOURCE[0]} contains the path of the script file in which that function is defined, ${FUNCNAME[1]} contains the name of the function from which the currently executing function was called, if applicable, and so on.

If a given function was invoked directly from the top-level scope in the script file that defined the function at level $i of the call stack, ${FUNCNAME[$i+1]} contains:

• main (a pseudo function name), if the script file was invoked directly (e.g., ./script)
• source (a pseudo function name), if the script file was sourced (e.g. source ./script or . ./script).