Is there an "in" operator in bash/bourne?

You can use case ... esac

$ cat in.sh 
#!/bin/bash

case "$1" in 
  "cat"|"dog"|"mouse")
    echo "dollar 1 is either a cat or a dog or a mouse"
  ;;
  *)
    echo "none of the above"
  ;;
esac

Ex.

$ ./in.sh dog
dollar 1 is either a cat or a dog or a mouse
$ ./in.sh hamster
none of the above

---

With ksh, bash -O extglob or zsh -o kshglob, you could also use an extended glob pattern:

if [[ "$1" = @(cat|dog|mouse) ]]; then
  echo "dollar 1 is either a cat or a dog or a mouse"
else
  echo "none of the above"
fi

With bash, ksh93 or zsh, you could also use a regular expression comparison:

if [[ "$1" =~ ^(cat|dog|mouse)$ ]]; then
  echo "dollar 1 is either a cat or a dog or a mouse"
else
  echo "none of the above"
fi

There is not an "in" test in bash, but there is a regex test (not in bourne):

if [[ $1 =~ ^(cat|dog|mouse)$ ]]; then
    echo "dollar 1 is either a cat or a dog or a mouse"
fi

And usually written using a variable (less problems with quoting):

regex='^(cat|dog|mouse)$'

if [[ $1 =~ $regex ]]; then
    echo "dollar 1 is either a cat or a dog or a mouse"
fi

For an older Bourne shell you need to use a case match:

case $1 in
    cat|dog|mouse)   echo "dollar 1 is either a cat or a dog or a mouse";;
esac

Using a case is fine and well if you have a fixed set of pets you want to match against. But it won't work if you need to build the pattern in runtime, as case doesn't interpret alternation from within expanded parameters.

This will match only the literal string cat|dog|mouse:

patt='cat|dog|mouse'
case $1 in 
        $patt) echo "$1 matches the case" ;; 
esac

You can, however, use a variable with the regular expression match. As long as the variable isn't quoted, any regex operators within it have their special meanings.

patt='cat|dog|mouse'
if [[ "$1" =~ ^($patt)$ ]]; then
        echo "$1 matches the pattern"
fi

You could also use associative arrays. Checking if a key exists in one is the closest thing to an in operator that Bash gives. Though the syntax is a bit ugly:

declare -A arr
arr[cat]=1
arr[dog]=1
arr[mouse]=1

if [ "${arr[$1]+x}" ]; then
        echo "$1 is in the array"
fi

(${arr[$1]+x} expands to x if arr[$1] is set, empty otherwise.)