For decades, it was believed that superfluid or superconducting properties of quasi-one-dimensional systems can be universally understood with a relatively simple quantum model known as Luttinger liquid. The discovery of the supertransport-through-solid effect in imperfect crystals of helium-4, always accompanied by even more striking effect of isochoric (i.e., fixed-volume) compressibility, has dramatically changed the situation. Based on first-principles simulations, there is little doubt that the unique behavior of imperfect helium-4 crystal is dictated by quasi-one-dimensional objects—edge dislocations with superfluid cores. Nevertheless, the unusual temperature dependence of the supertransport effect does not fit into the Luttinger liquid paradigm. I will explain that here, and in a few other instructive setups, we are dealing with a fundamentally new type of quasi-one-dimensional superfluids, the so-called transverse quantum fluids. From a fundamental point of view, transverse quantum fluids provide us with an exciting demonstration of conditional character of many dogmas associated with superfluidity.