I recall seeing the answer to this quandary before, but I think that quandary was made on the official forums before they died a horrible, horrible death at the hands of their creator. And I can't remember what the answer was. So:

If somebody has a Humility out and casts a face-down creature, can its Morph ability be used?

I ask this because recently in a Commander game with friends somebody played a Humility while somebody else was tapped out with a face-down creature on the battlefield and everybody was trying to figure out if he could turn it face-up. Being the go-to rules guy in my playgroup, everybody turned to me for the answer. I told them truthfully that I wasn't 100% sure on the matter, but that I was mostly sure (78%) that the face-down creature would be a 1/1 with no abilities, including the morph ability on its face. I went on to explain that Manifest wouldn't be affected in the same manner, since it's part of the Manifest rules (as opposed to something on the card) that's allowing you to turn the permanent face-up for its mana cost.

I would say that you could morph the creature, but that it would flip over as a 1/1 with no abilities so if you were trying to get something cool out of morphing it, you are out of luck.

Right. To unmorph the creature, first you have to reveal it and determine what its face up characteristics would be so that you can pay its morph cost. Due to Humility, however, you determine that the face up creature wouldn't have morph and so you are unable to pay its morph cost to turn it face up.

702.36e. Any time you have priority, you may turn a face-down permanent you control with a morph ability face up. This is a special action; it doesn't use the stack (see rule 115). To do this, show all players what the permanent's morph cost would be if it were face up, pay that cost, then turn the permanent face up. (If the permanent wouldn't have a morph cost if it were face up, it can't be turned face up this way.) The morph effect on it ends, and it regains its normal characteristics. Any abilities relating to the permanent entering the battlefield don't trigger when it's turned face up and don't have any effect, because the permanent has already entered the battlefield.

The person with the morph creature said he had a Mastery of the Unseen in his deck, so it could have become relevant. Hence why I mentioned the lack of interaction between Humility and Manifest.

You are also correct that players can still turn manifested cards face-up, since that's not an ability on the card.

As an additional note: noncreature morphs like Zoetic Cavern can be turned face up in that scenario, because they're unaffected by Humility once they're face-up.