POSIX catch newline in case statement

The dash shell does not have C-strings ($'...'). C-strings is an extension to the POSIX standard. You would have to use a literal newline. This is easier (and looks nicer) if you store the newline in a variable:

#!/bin/dash

nl='
'

for string; do

    case $string in
        *"$nl"*)
            printf '"%s" contains newline\n' "$string"
            ;;
        *)
            printf '"%s" does not contain newline\n' "$string"
    esac

done

For each command line argument given to the script, this detects whether it contains a newline or not. The variable used in the case statement ($string) does not need quoting, and the ;; after the last case label is not needed.

Testing (from an interactive zsh shell, which is where the dquote> secondary prompt comes from):

$ dash script.sh "hello world" "hello
dquote> world"
"hello world" does not contain newline
"hello
world" contains newline

You can include a literal newline (in quotes) as the pattern, just as you did in assigning to the variable:

q='
'
case "$q" in
    *'
'*) echo nl;;
    *) echo NO nl;;
esac

This makes the formatting ugly (you cannot indent the end quote), but should be fully portable. I tested in bash, zsh, ksh, and dash.