See below for the script used in these tests.
Grep can only operate on stdin, so therefore you must convert the stderr stream in a form that Grep can parse.
Normally, output is printed to both stdout and stderr:
$ ./stdout-stderr.sh
./stdout-stderr.sh: Printing to stdout
./stdout-stderr.sh: Printing to stderr
So we hide output to stdout:
$ { ./stdout-stderr.sh >/dev/null ; } 2>&1
./stdout-stderr.sh: Printing to stderr
The following Bash syntax will hide output to stdout, but will still show stderr. First we pipe stdout to /dev/null, then we convert stderr to stdout, because Unix pipes will only operate on stdout. You can still grep the text.
$ { ./stdout-stderr.sh >/dev/null ; } 2>&1 | grep err
./stdout-stderr.sh: Printing to stderr
Here's the script I use for testing. This prints one line to stdout and one line to stderr:
#!/bin/sh
# Print a message to stdout
echo "$0: Printing to stdout"
# Print a message to stderr
echo "$0: Printing to stderr" >&2
exit 0