In which conflicts has dogfighting played a role of any significance in aerial warfare?
Korea had the infamous "MiG Alley" and included the first jet-to-jet kills, but the Vietnam war was more of the end of dogfighting as we know it (close combat, cannon kills.) The USAF switched to primarily missile tactics where a pilot would get a radar lock, fire his missiles from miles away, and leave the area. The F-4 Phantom was the first American pursuit/fighter aircraft since WWI to not have a cannon built in somehow because of this new change of doctrine. The MiG pilots opposing them (some Soviet, some Chinese, some Vietnamese) would often close within missile range and tangle with the opposing, virtually unarmed at this point, fighters. With a raising loss rate, a gun pod was added to the Phantoms and dogfighting training was added - the famous Top Gun school. Because of lessons learned in Vietnam, all modern USAF fighters/interceptors have a cannon for defensive and offensive purposes, while missile kills are still preferred.
With American air superiority being key to success, more strategic bombing and missile strikes on airfields allows for American fighters to not need to attempt dogfighting, as there are rarely any opposing forces in the air to dogfight. As glorified as it remains, dogfighting is becoming very uncommon. One of the last air-to-air kills by a US aircraft was when an F-15E Strike Eagle in ground support configuration bombed a Hind helicopter, as recounted here.