A sample of large northern Spitzer Infrared Nearby Galaxies Survey (SINGS) galaxies has recently been observed with the Westerbork Synthesis Radio Telescope (WSRT). We present observations of the linearly polarized radio continuum emission in this WSRT-SINGS galaxy sample. Of the 28 galaxies treated in this paper, 21 are detected in polarized radio continuum at 18- and 22-cm wavelengths. We utilize the rotation measure synthesis (RM-Synthesis) method, as implemented by Brentjens & de Bruyn, to coherently detect polarized emission from a large fractional bandwidth, while simultaneously assessing the degree of Faraday rotation experienced by the radiation along each line-of-sight. This represents the first time that the polarized emission and its Faraday rotation have been systematically probed down to ~10 microJy/beam RMS for a large sample of galaxies. Non-zero Faraday rotation is found to be ubiquitous in all of the target fields, from both the Galactic foreground and the target galaxies themselves. In this paper, we present an overview of the polarized emission detected in each of the WSRT-SINGS galaxies. The most prominent trend is a systematic modulation of the polarized intensity with galactic azimuth, such that a global minimum in the polarized intensity is seen toward the kinematically receding major axis. The implied large-scale magnetic field geometry is discussed in a companion paper. A second novel result is the detection of multiple nuclear Faraday depth components that are offset to both positive and negative RM by 100-200 rad/m^2 in all targets that host polarized (circum-)nuclear emission.
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