Voidmage Apprentice is a creature with Morph that counters target spell when it is turned face up.
I've read that it can counter spells with split second. How does it do this? Is it that the ability is an added effect of the morph action and does not use the stack? Or is it a triggered ability that split second cannot prevent from being placed on the stack?
The Gatherer ruling on Voidmage Apprentice explains it fairly well:
If a spell with split second is on the stack, you can still respond by turning this creature face up and targeting that spell with the trigger. This is because split second only stops players from casting spells or activating abilities, while turning a creature face up is a special action.
The triggered ability still does use the stack, but split second doesn't stop it. And of course, you're able to turn it face up in the first place because that's a special action, not an activated ability that uses the stack.
Unmorphing a card does not use the stack. This is how it gets around the split second effect.
This behavior is very similar to mana source behaviors of creatures as discussed in this recent post: Does killing a creature mana source that's being tapped for mana to pay for a spell counter that spell?
The reason this behavior exists, is it prohibits your opponent from smoking your morphling with a lightning bolt when you declare you wish to unmorph. It makes morphing more useful.
Here's the rule for unmorphing, that shows it doesn't use the stack.
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). ...[stuff deleted]...
