There is almost always an environment variable $SHELL that you can learn this from. There are some edge cases where this will fail, particularly if one interactive shell is used to launch another. Most recent shells also set a variable about themselves in the form of $BASH_VERSION or $ZSH_VERSION. I don't know that sh does this.
A bit fancier:
Alternatively, most interactive shells will know what they are and give you some useful output to if you run something like echo $0. This appears to rely on about the same information as you would get if you ran ps -fp $$, which would retrieve the process data for the process that launched ps.
A way that might work for some shells and environments that don't have convenience things such as environment variables set would be to rip the data you want out of proc and figure out what the running executable path is actually pointing to using readlink -f /proc/$$/exe. This has the advantage of giving you information about what executable was lauched even if it was a symlink to something else, in which case the SHELL variable might be lieing to you.