B733's 4's, 737's & 738's can all do Flap 1 takeoffs. I've not flown 5's or 6's.
What has happened on a F1 takeoff is at the call "bug up", (to accelerate the a/c for flap retraction) the RHS pilot, who might nave been changing frequency and chatting to ATC during departure, has heard the word UP and selected Flaps UP. Scary, but true.