Measure each tear with measuring tape. Cut a piece of canvas to the dimensions of each tear, making each patch slightly bigger than each tear.
Slide each patch of canvas behind each tear and smooth it out. Apply a thin layer of high-grade contact cement on the visible part of the canvas. Press the upper layer of torn vinyl against the cement, smoothing the tear down.
Apply a thin layer of leather repair glue -- in a color that matches your vinyl -- around where the edges of the tear meet the surrounding vinyl.
Cover the glue with grain paper that matches your vinyl couch. For one minute, press a hot iron over the paper to set the glue in place. Remove the iron and peel off the paper.