How to determine if a string is a substring of another in bash?

[[ "bcd" =~ "ab" ]]
[[ "abc" =~ "ab" ]]

the brackets are for the test, and as it is double brackets, it can so some extra tests like =~.

So you could use this form something like

var1="ab"
var2="bcd"
if [[ "$var2" =~ "$var1" ]]; then
    echo "pass"
else
    echo "fail"
fi

Edit: corrected "=~", had flipped.

You can use the form ${VAR/subs} where VAR contains the bigger string and subs is the substring your are trying to find:

my_string=abc
substring=ab
if [ "${my_string/$substring}" = "$my_string" ] ; then
  echo "${substring} is not in ${my_string}"
else
  echo "${substring} was found in ${my_string}"
fi

This works because ${VAR/subs} is equal to $VAR but with the first occurrence of the string subs removed, in particular if $VAR does not contains the word subs it won't be modified.

Using bash filename patterns (aka "glob" patterns)

substr=ab
[[ abc == *"$substr"* ]] && echo yes || echo no    # yes
[[ bcd == *"$substr"* ]] && echo yes || echo no    # no