  Even when ideal solids are insulating, their states with crystallographic
defects may have superfluid properties. It became clear recently that edge
dislocations in $^4$He featuring a combination of microscopic quantum roughness
and superfluidity of their cores may represent a new paradigmatic class of
quasi-one-dimensional superfluids. The new state of matter, termed transverse
quantum fluid (TQF), is found in a variety of physical setups. The key
ingredient defining the class of TQF systems is infinite compressibility, which
is responsible for all other unusual properties such as the quadratic spectrum
(or even the absence) of normal modes, irrelevance of the Landau criterion,
off-diagonal long-range order at $T = 0$, and the exponential dependence of the
phase slip probability on the inverse flow velocity. From a conceptual point of
view, the TQF state is a striking demonstration of the conditional character of
many dogmas associated with superfluidity, including the necessity of
elementary excitations, in general, and the ones obeying Landau criterion in
particular.
