Is $\{a^nb^m \mid n,m\ge 0, n\ne m\}$ regular or not?
Using Ran G.'s comment which links to a very similar question:

Suppose $L$ is regular. $M=a^*b^*$ is regular, too, so since regularity is closed by set difference, $M\backslash L=\{a^nb^n\}$ would be, too, which is the usual counterexample.