  We draw attention to the extreme filamentary structures seen in
high-resolution optical images of certain planetary nebulae. We determine the
physical properties of the filaments in the nebulae IC 418, NGC 3132, and NGC
6537, and based on their large length-to-width ratios, longitudinal coherence,
and morphology, we suggest that they may be signatures of the underlying
magnetic field. The fields needed for the coherence of the filaments are
probably consistent with those measured in the precursor circumstellar
envelopes. The filaments suggest that magnetic fields in planetary nebulae may
have a localized and thread-like geometry.
