We present a study of quasar selection using the supernova fields of the Dark Energy Survey (DES). We used a quasar catalog from an overlapping portion of the SDSS Stripe 82 region to quantify the completeness and efficiency of selection methods involving color, probabilistic modeling, variability, and combinations of color/probabilistic modeling with variability. In all cases, we considered only objects that appear as point sources in the DES images. We examine color selection methods based on the Wide-field Infrared Survey Explorer (WISE) mid-IR W1 - W2 color, a mixture of WISE and DES colors (g - i and i - W1), and a mixture of Vista Hemisphere Survey and DES colors (g - i and i - K). For probabilistic quasar selection, we used XDQSO, an algorithm that employs an empirical multi-wavelength flux model of quasars to assign quasar probabilities. Our variability selection uses the multi-band chi(2)-probability that sources are constant in the DES Year 1 griz-band light curves. The completeness and efficiency are calculated relative to an underlying sample of point sources that are detected in the required selection bands and pass our data quality and photometric error cuts. We conduct our analyses at two magnitude limits, i < 19.8 mag and i < 22 mag. For the subset of sources with W1 and W2 detections, the W1 - W2 color or XDQSOz method combined with variability gives the highest completenesses of > 85% for both i-band magnitude limits and efficiencies of >80% to the bright limit and > 60% to the faint limit; however, the giW1 and giW1+ variability methods give the highest quasar surface densities. The XDQSOz method and combinations of W1W2/giW1/XDQSOz with variability are among the better selection methods when both high completeness and high efficiency are desired. We also present the OzDES Quasar Catalog of 1263 spectroscopically confirmed quasars from three years of OzDES observation in the 30 deg(2) of the DES supernova fields. The catalog includes quasars with redshifts up to z similar to 4 and brighter than i = 22 mag, although the catalog is not complete up to this magnitude limit.