I am writing a set of scripts that I want to be portable, but I need to know whether sh on the current platform stands for bash, ksh, or ash. Is there a clear way to do it?

What comes to my mind first is to inspect which shell has which --version:

$ zsh --version
zsh 5.0.2 (x86_64-apple-darwin13.0)

$ bash --version
GNU bash, version 4.3.39(1)-release (x86_64-apple-darwin13.4.0)
Copyright (C) 2013 Free Software Foundation, Inc.
License GPLv3+: GNU GPL version 3 or later <http://gnu.org/licenses/gpl.html>

This is free software; you are free to change and redistribute it.
There is NO WARRANTY, to the extent permitted by law.

$ ksh --version
  version         sh (AT&T Research) 93u+ 2012-08-01

$ dash --version
dash: 0: Illegal option --

$ pdksh --version
pdksh: pdksh: --: unknown option

Apart from being clumsy, this doesn't even produce results in all cases.

Edit

I need it for my bashrc/zshrc/...-like project, where I assign my current working shell to a variable, and use that variable everywhere.

I can't post my code because I need to solve the problem to enable overall cleanliness of my work. Moreover, it would be too much monkeycode... don't misunderstand it, but POSIX compatibility is too narrow to make my project small enough. I'd need to crutch on system configs otherwise.

However, I can post my UNIX Shell defining function:

PROJ_GET_SHELL () {
    local PROJ_SHELL="`ps -p $$ | tail -1 | tr ' ' '\n' | tail -1`"
    PROJ_local=(pdksh bash dash mksh zsh ksh sh)
    for i in ${PROJ_local[*]}
    do
        if ! [ -z `echo $PROJ_SHELL | grep $i` ]
        then
            echo "$i"
            break
        fi
    done
}

PS. The least bit of research shows that $SHELL doesn't change when running a shell as subprocess.