The excellent properties of elastomers are exploited to trigger wrinkling instabilities in curved shells. Micro‐ and nano‐fibres are produced by electrospinning and UV irradiated: each fibre consists of a soft core and a stiff outer half‐shell. Upon solvent de‐swelling, the fibres curl because the shell and the core have different natural lengths. Wrinkling only starts after the fibre has attained a well‐defined helical shape. A simple analytical model is proposed to find the curling curvature and wrinkle wavelength, as well as the transition between the “curling” and “wrinkling” regimes. This new instability resembles that found in the tendrils of climbing plants as they dry and lignify. image