The Australian SKA Pathfinder (ASKAP) radio telescope has carried out a survey of the entire Southern Sky at 887.5 MHz. The wide area, high angular resolution, and broad bandwidth provided by the low-band Rapid ASKAP Continuum Survey (RACS-low) allow the production of a next-generation rotation measure (RM) grid across the entire Southern Sky. Here we introduce this project as Spectral and Polarisation in Cutouts of Extragalactic sources from RACS (SPICE-RACS). In our first data release, we image 30 RACS-low fields in Stokes I, Q, U at 25$^{\prime\prime}$ angular resolution, across 744–1032 MHz with 1 MHz spectral resolution. Using a bespoke, highly parallelised, software pipeline we are able to rapidly process wide-area spectro-polarimetric ASKAP observations. Notably, we use ‘postage stamp’ cutouts to assess the polarisation properties of 105912 radio components detected in total intensity. We find that our Stokes Q and U images have an rms noise of $\sim$80 $\unicode{x03BC}$Jy PSF$^{-1}$, and our correction for instrumental polarisation leakage allows us to characterise components with $\gtrsim$1% polarisation fraction over most of the field of view. We produce a broadband polarised radio component catalogue that contains 5818 RM measurements over an area of $\sim$1300 deg$^{2}$ with an average error in RM of $1.6^{+1.1}_{-1.0}$ rad m$^{-2}$, and an average linear polarisation fraction $3.4^{+3.0}_{-1.6}$ %. We determine this subset of components using the conditions that the polarised signal-to-noise ratio is $>$8, the polarisation fraction is above our estimated polarised leakage, and the Stokes I spectrum has a reliable model. Our catalogue provides an areal density of $4\pm2$ RMs deg$^{-2}$; an increase of $\sim$4 times over the previous state-of-the-art (Taylor, Stil, Sunstrum 2009, ApJ, 702, 1230). Meaning that, having used just 3% of the RACS-low sky area, we have produced the 3rd largest RM catalogue to date. This catalogue has broad applications for studying astrophysical magnetic fields; notably revealing remarkable structure in the Galactic RM sky. We will explore this Galactic structure in a follow-up paper. We will also apply the techniques described here to produce an all-Southern-sky RM catalogue from RACS observations. Finally, we make our catalogue, spectra, images, and processing pipeline publicly available.