Premixed-gas flames subject to steady spatially-varying straining flows were studied to examine one aspect of premixed flames in strongly turbulent flows where strain rate gradients are present and local strain rates may be high enough to cause local flame-front extinguishment. The spatially-varying straining flows were created using a counterflow slot-jet burner with slightly non-parallel jet exits. When the flow configuration was premixed combustible gas vs. cold inert gas, so that only a single flame was produced, steady flame "edges" could be created where the flame would exist in the low-strain region but would be extinguished in the high-strain region. When the flow configuration was premixed gas vs. premixed gas, twin flames would exist in the low-strain region that converged to a corner-like tip in the high-strain region. For both configurations the local strain at the location of the stationary flame edge was somewhat lower than the strain required to extinguish flames in the same mixture subject to a spatially uniform strain. The difference was greater for the twin-flame configuration, particularly at high Lewis number (Le). Due to diffusive-thermal instabilities, cellular flames were observed at low Le and travelling-wave patterns were observed at high Le. Le effects also led to the formation of isolated "flame tubes" rather than continuous fronts at sufficiently low Le and high strain rates. All of these results are consistent with recent theoretical predictions. These results indicate that "laminar flamelet" models of premixed turbulent combustion may be reasonably accurate for single flames over a wide range of Le and twin flames with Le close to unity, even at conditions approaching those where local flame quenching occurs, but may not be accurate for twin-flames except for Le near unity. This finding is somewhat different from previous experimental and theoretical results for nonpremixed edge-flames, where more substantial differences between uniform flames and edge-flames were found for all Le.