We measure the polarization fraction of a sample of 6282 Galactic cold clumps at 353 GHz, consisting of Planck Galactic cold clump (PGCC) catalogue category 1 objects (flux densities measured with signal-to-noise ratio (S/N) > 4). We find the mean-squared polarization fraction at 353 GHz to be Π 2 = [4.79 ± 0.44] × 10 −4 equating to an 11 𝜎 detection of polarization. We test if the polarization fraction depends on the clumps' physical properties, including flux density, luminosity, Galactic latitude and physical distance. We see a trend towards increasing polarization fraction with increasing Galactic latitude, but find no evidence that polarization depends on the other tested properties. The Simons Observatory, with angular resolution of order 1 and noise levels between 22 and 54 𝜇K−arcmin at high frequencies, will substantially enhance our ability to determine the magnetic field structure in Galactic cold clumps. At ≥ 5 𝜎 significance, we predict the Simons Observatory will detect at least ∼12,000 cold clumps in intensity and ∼430 cold clumps in polarization. This number of polarization detections would represent a two orders of magnitude increase over the current Planck results.