A Pulse Detonation Engine was proposed for DARPA's Blackswift hypersonic test vehicle from the late 00's. Pulse Detonation Engines are engines that use the same detonation concept but are still inherently air breathing jets and not rockets. I'm not aware of if one was ever built though. It's my understanding that contemporary hypersonic missiles use solid rocket motors to get up to speed and then glide to their target.