You want a "negative lookahead":

use warnings;
use strict;

while(<DATA>){
    print if /
                cat # word cat
                (?: # do not capture
                (?! # negative lookahead
                \s* # possible whitespace
                dog # the word dog
                )   # end negative lookahead
                )   # end do not capture group
            /x;

    # /cat(?:(?!\s*dog))/
}

__DATA__
my cat loves dogs
catdog
cat dog dog dog
dog cat dog
dog cat not a dog
[download]

Output:

my cat loves dogs
dog cat not a dog
[download]

Note that the above will also match things like cat doggle, so if you *only* want it to match the word "dog", you'll have to put a boundary at the end of that word in the regex.

Update: AnomalousMonk pointed out below that the non-capture portion is not required here. For posterity, I'll leave it in (unless someone advises it'd be best to remove). So essentially, the second and last lines in the regex above are not needed.