Laughing at your own mistake out loud gives everyone else permission to stop being awkward about it.

Laughing first at your own mistake lets everybody else off the hook. A room that has just watched you drop a tray or lose your place is holding its breath, unsure whether to look away, and your reaction decides which. Once you treat it as survivable, so does everyone, and the moment is over in about four seconds.

Say something out loud the next time you fumble in front of people, even just that it was a bad one. Then carry on with what you were doing. Do it without the apology, because sorry keeps the moment open and a laugh closes it.