For example; if a third level Arcane Archer fighter chose to fire a 'Bursting Shot' and missed the target, is one of his uses of the Arcane Shot class feature expended?
(Xanathar's Guide to Everything, page 28)
Under Arcane shot, when first describing the feature it says:
Once per turn when you fire an arrow from a shortbow or longbow as part of the Attack action, you can apply one of your Arcane Shot options to that arrow. You decide to use the option when the arrow hits a creature, unless the option doesn’t involve an attack roll. You have two uses of this ability, and you regain all expended uses of it when you finish a short or long rest.
And, it says:
You imbue your arrow with force energy drawn from the school of evocation. The energy detonates after your attack. Immediately after the arrow hits the creature, the target and all other creatures within 10 feet of it take 2d6 force damage each.
So I don't think you use the feature until you know that attack was successful, so you wouldn't lose the feature on a miss.