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ABSTRACT The early phases of the observed evolution of the supernovae (SNe) are expected to be dominated by the shock breakout and ‘flash’ ionization of the surrounding circumstellar medium. This material arises from the last stages of the evolution of the progenitor, such that photometry and spectroscopy of SNe at early times can place vital constraints on the latest and fastest evolutionary phases leading up to stellar death. These signatures are erased by the expansion of the ejecta within ∼5 d after explosion. Here we present the earliest constraints, to date, on the polarization of 10 transients discovered by the Zwicky Transient Facility (ZTF), between 2018 June and 2019 August. Rapid polarimetric follow-up was conducted using the Liverpool Telescope RINGO3 instrument, including three SNe observed within <1 d of detection by the ZTF. The limits on the polarization within the first 5 d of explosion, for all SN types, is generally $\lt 2{ per\ cent}$, implying early asymmetries are limited to axial ratios >0.65 (assuming an oblate spheroidal configuration). We also present polarimetric observations of the Type I Superluminous SN 2018bsz and Type II SN 2018hna, observed around and after maximum light.