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Within the six-parameter nonlinear shell theory we analyzed the in-plane rotational instability which occurs under in-plane tensile loading. For plane deformations the considered shell model coincides up to notations with the geometrically nonlinear Cosserat continuum under plane stress conditions. So we considered here both large translations and rotations. The constitutive relations contain some additional micropolar parameters with so-called coupling factor that relates Cosserat shear modulus with the Cauchy shear modulus. The discussed instability relates to the bifurcation from the static solution without rotations to solution with non-zero rotations. So we call it rotational instability. We present an elementary discrete model which captures the rotational instability phenomenon and the results of numerical analysis within the shell model. The dependence of the bifurcation condition on the micropolar material parameters is discussed. [ABSTRACT FROM AUTHOR]