For isolated vacancies in ordered local-moment antiferromagnets we show that the magnetic-field linear-response limit is generically singular: The magnetic moment associated with a vacancy in zero field is different from that in a finite field h in the limit h→0 ⁺. The origin is a universal and singular screening cloud, which moreover leads to perfect screening as h→0 ⁺ for magnets which display spin-flop bulk states in the weak-field limit.