  Impurities act as in situ probes of nontrivial electronic structure, causing
real-space modulations in the density of states detected by scanning tunneling
spectroscopy on the sample surface. We show that distinctive topological
features of Weyl semimetals can be revealed in the Fourier transform of this
map, interpreted in terms of quasiparticle interference (QPI). We develop an
exact Green's function formalism and apply it to generalized models of Weyl
semimetals with an explicit surface. The type of perturbation lifting the Dirac
node degeneracy to produce the three-dimensional bulk Weyl phase determines the
specific QPI signatures appearing on the surface. QPI Fermi arcs may or may not
appear, depending on the relative surface orientation and quantum interference
effects. Line nodes give rise to tube projections of width controlled by the
bias voltage. We consider the effect of crystal warping, distinguishing
dispersive arclike features from true Fermi arcs. Finally, we demonstrate that
the commonly used joint-density-of-states approach fails qualitatively, and
cannot describe QPI extinction.
