We present high spatial resolution integral-field spectroscopy of 28 elliptical (E) and lenticular (S0) galaxies from the SAURON representative survey obtained with the OASIS spectrograph during its operation at the Canada-France-Hawaii Telescope. These seeing-limited observations explore the central 8 × 10 arcsec2 (typically 1 kpc diameter) regions of these galaxies using a spatial sampling four times higher than SAURON (0.27-arcsec versus 0.94-arcsec spatial elements), resulting in almost a factor of 2 improvement in the median point spread function. These data allow accurate study of the central regions to complement the large-scale view provided by SAURON. Here we present the stellar and gas kinematics, stellar absorption-line strengths and nebular emission-line strengths for this sample. We also characterize the stellar velocity maps using the `kinemetry' technique, and derive maps of the luminosity-weighted stellar age, metallicity and abundance ratio via stellar population models. We give a brief review of the structures found in our maps, linking also to larger-scale structures measured with SAURON. We present two previously unreported kinematically decoupled components (KDCs) in the centres of NGC3032 and NGC4382. We compare the intrinsic size and luminosity-weighted stellar age of all the visible KDCs in the full SAURON sample, and find two types of components: kiloparsec-scale KDCs, which are older than 8 Gyr, and are found in galaxies with little net rotation; and compact KDCs, which have intrinsic diameters of less than a few hundred parsec, show a range of stellar ages from 0.5 to 15 Gyr (with 5/6 younger than 5 Gyr), are found exclusively in fast-rotating galaxies, and are close to counter-rotating around the same axis as their host. Of the seven galaxies in the SAURON sample with integrated luminosity-weighted ages less than 5 Gyr, five show such compact KDCs, suggesting a link between counter-rotation and recent star formation. We show that this may be due to a combination of small sample size at young ages, and an observational bias, since young KDCs are easier to detect than their older and/or corotating counterparts. Based on observations obtained at the Canada-France-Hawaii Telescope which is operated by the National Research Council of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii. E-mail: mcdermid [at] strw.leidenuniv.nl (mcdermid[at]strw[dot]leidenuniv[dot]nl)