We present the first systematic spectropolarimetric study of Luminous Blue Variables (LBVs) in the Galaxy and the Magellanic Clouds, in order to investigate the geometries of their winds. We find that at least half of our sample show changes in polarization across the strong H$alpha$ emission line, indicating that the light from the stars is intrinsically polarized and therefore that asphericity already exists at the base of the wind. Multi-epoch spectropolarimetry on four targets reveals variability in their intrinsic polarization. Three of these, AG Car, HR Car and P Cyg, show a position angle (PA) of polarization which appears random with time. Such behaviour can be explained by the presence of strong wind-inhomogeneities, or `clumps within the wind. Only one star, R 127, shows variability at a constant PA, and hence evidence for axi-symmetry as well as clumpiness. However, if viewed at low inclination, and at limited temporal sampling, such a wind would produce a seemingly random polarization of the type observed in the other three stars. Time-resolved spectropolarimetric monitoring of LBVs is therefore required to determine if LBV winds are axi-symmetric in general. The high fraction of LBVs ($>$ 50%) showing intrinsic polarization is to be compared with the lower $sim$ 20-25 % for similar studies of their evolutionary neighbours, O supergiants and Wolf-Rayet stars. We anticipate that this higher incidence is due to the lower effective gravities of the LBVs, coupled with their variable temperatures within the bi-stability jump regime. This is also consistent with the higher incidence of wind asphericity that we find in LBVs with strong H$alpha$ emission and recent (last $sim$ 10 years) strong variability.
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