What happens to an Aura if its enchanted permanent loses all abilities?

An Aura enchanting a permanent remains attached even if that permanent loses all abilities. The rule governing Auras staying attached only checks whether the permanent is a legal object to be enchanted (i.e., it still exists on the battlefield and meets the Aura's enchant condition), not whether it has any particular abilities (CR 704.5m, 303.4).

However, there is an important nuance: if the permanent loses all abilities, any abilities granted to it by the Aura (such as flying or trample) are also removed as part of losing all abilities. The Aura card still exists and is still attached, but its effect of granting abilities is overridden by the loss-of-abilities effect, depending on timestamp and layer ordering (CR 613, Layer 6).

The Aura's own static abilities (like those that grant the enchanted creature +2/+2) still exist on the Aura card itself. They simply may not be producing any visible effect on a creature that has lost all abilities, because granting abilities happens in Layer 6 and losing all abilities from an effect like Humility also operates in that layer — the one with the later timestamp wins (CR 613.7).

Concrete example: Your opponent's Humility ("All creatures lose all abilities and are 1/1") is already in play. You then enchant your creature with Rancor. Because Humility has an earlier timestamp and both effects apply in Layer 6, Rancor's trample-granting ability is overridden by Humility — your creature won't have trample. But Rancor remains attached to the creature and is not put into a graveyard.

In summary: the Aura stays on the permanent regardless of that permanent losing abilities (CR 303.4c), but whether the Aura's granted abilities are actually functioning depends on layer and timestamp interaction (CR 613.7).