We report on strain engineering of superconductivity in RuO_{2} single-crystal films, which are epitaxially grown on rutile TiO_{2} and MgF_{2} substrates with various crystal orientations. Systematic mappings between the superconducting transition temperature and the lattice parameters reveal that shortening of specific ruthenium-oxygen bonds is a common feature among the superconducting RuO_{2} films. Ab initio calculations of electronic and phononic structures for the strained RuO_{2} films suggest the importance of soft phonon modes for emergence of the superconductivity. The findings indicate that simple transition metal oxides such as those with a rutile structure may be suitable for further exploring superconductivity by controlling phonon modes through the epitaxial strain.