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Rheological studies of low concentration (down to 0.1%) poly(acrylonitrile) (PAN)–dimethyl sulfoxide (DMSO) solutions revealed the existence of a rather unusual effect of the gel-like structure formation, while this phenomenon is absent at higher concentrations (17–24 wt %) of PAN in DMSO. The evidence of the gel-like structure consists in the pronounced yielding, which can be detected in the 0.1–5% concentration range and plateau on the frequency dependence of the storage modulus. In addition, rather large particles were found by the dynamic light scattering method. The gel-like structure of dilute PAN solutions is similar to supramolecular gels, while the gelation of water containing systems brings to mind a phase transition. The observed maximal Newtonian viscosity at the low shear stresses for dilute solutions should be treated as an artifact related to the method of measuring, and at long-term loading the plateau is converted to the yield stress. Dilute PAN/DMSO gels are thixotropic. A possibility of bifurcation (due to coexistence of different states) leads to self-oscillations of the stress response in a wide shear rate region. The dynamic viscosity coincides with the shear apparent viscosity for high concentrations but does not in the low concentration range. Such a kind of rheological behavior is not observed if dimethylformamide or aqueous solution of sodium thiocyanate was used instead of DMSO as a PAN solvent. Possible chemical structures responsible for the gel structure formation are discussed.