Mahlo operation, consistency border Can a (relatively consistent) cardinal notion be given so that its usual Mahlo operation
is (probably at least) not consistent?

Sure - "is a successor ordinal." For any (fine, uncountable) infinite cardinal $\kappa$, the set of limit cardinals below $\kappa$ is a club which avoids the set of successor ordinals.
I suspect, though, that this isn't really what you want. So in the opposite direction, let me observe:

There is no known, currently-considered-consistent large cardinal property whose Mahlo version is known to be (or even suspected to be) inconsistent.

This is an awkward thing to claim since it's inherently hard (if not impossible) to justify, but it is to the best of my knowledge true.
