  We prove a Painlev\'e theorem for bounded quasiregular curves in Euclidean
spaces extending removability results for quasiregular mappings due to Iwaniec
and Martin. The theorem is proved by extending a fundamental inequality for
volume forms to calibrations and proving a Caccioppoli inequality for
quasiregular curves.
  We also establish a qualitatively sharp removability theorem for quasiregular
curves whose target is a complete Riemannian manifold with sectional curvature
bounded from above. As an application, we extend a theorem of Bonk and Heinonen
for quasiregular mappings to the setting of quasiregular curves: every
non-constant quasiregular $\omega$-curve from $\mathbb{R}^n$ into $( N, \omega
)$, where the bounded cohomology class of $\omega$ is in the bounded K\"unneth
ideal, has infinite energy.
