My favourites:

-blink (short range teleportation)
-slowfall/glide (like LoZ: Windwakers leaf, or the double jump ability for the sorceress in Orcs must Die 2)
-(double) hookshot

Though shapeshifting is cool to. and it allows for many different abilities to given within a more coherent theme.

-(double) hookshot

What is a double-hookshot?

[Edit:] Oh, right, I had forgot about these. Thank you for reminding me!

- Compass generally pointing into the direction of "the good stuff" (or feeling drawn towards it)

- (with time slowing) catch (many) falling objects, eg keys.

- Sensing danger

- Wizard stuff to you, instead of having to fetch it (get key or bottle across the room to you, or fetch boat from the other side of the river)

- Summon animals to look, or bring you things, or be used as bridge (mr Bond escaping from a crocodile island comes to mind), people being brought by birds.

Edit:

- Cook magical meal to get secrets from a general (a truth-telling potion would also work I guess).

- Disappear in the scenery to evade guards (morphing ability)

- Phase out of reality, becoming unobservable (guards, motion detectors)

- Figure out timing of guards to sneak through

- Disable or control electrical equipment

- Anti gravity boots / walk on air, eg avoiding seismic ground sensors

I'm trying to think of ways other than "Do you have X ability? Cool, door unlocked!"

I've been thinking that it would be neat if there were a game in which world-unlocks were done entirely through knowledge. No hard ability-gates (in the sense that you couldn't possibly pass until gaining the ability), just information about the possibilities of your character and the world that you wouldn't have at the beginning of the game.

For example, say your character is an ethnobotanist exploring an ancient jungle temple. There are various kinds of plants growing everywhere, and seeds lying around, which can help you traverse the world if you know how to utilize them (water this one, pollinate that one, germinate this seed only in the sunlight and that one only in the shade, etc.), but there's an element of randomness to it and the combinatoric possibilities are rather larger than you would necessarily want to brute-force. The "ability unlocks" in the game wouldn't be like a new pair of shoes that give you a double jump; they'd just be information that you don't yet have. (Like you'd find a carving in the wall that makes clear that watering the plant with star-shaped leaves creates a climbable vine in a particular direction. If you didn't already know that, that unlocks more of the world.)

Or you're an explorer in an abandoned alien station, slowly learning what actions power the machinery and make it affect the world. Or you're a wizard learning the combinatoric properties of your rune tiles. Something like that.

This kind of system has two properties that I find interesting. The first is that there's an additional possibility if you get stuck or lost, which is just plain experimentation. There's never a hard gate. (I find it less interesting if the system is so complicated and random you can never experiment or guess. Then it'd essentially be a gate locked with a password, which is nothing interesting or special.)

The other is that on subsequent playthroughs, you're in an interesting situation. You don't know exactly what to do at each point (because of the element of randomization), but on the other hand you're not a total beginner. You know the basic logical structure of the upgrade system, and so can probably deduce some missing pieces of the system based on what you've seen so far. ("So I know that no two plants are activated the same way, and no two activate-able plants ever occur next to each other. There's a gap here that I can probably cross, and only one plant that isn't ever next to a plant I know. I already know a plant that activates with salt water, and this one's too far away from a light source to be the light-activated plant, so...") Each time you play you can probably get through more quickly, since you can deduce more about the system from each clue you get.

I'm trying to think of ways other than "Do you have X ability? Cool, door unlocked!"

I've been thinking that it would be neat if there were a game in which world-unlocks were done entirely through knowledge. No hard ability-gates (in the sense that you couldn't possibly pass until gaining the ability), just information about the possibilities of your character and the world that you wouldn't have at the beginning of the game.

Yeah, that's an awesome way of exploring a world.

In fact, the soon-to-be-released AAA-indie game "The Witness" (by the same developer who made Braid) is entirely built around this idea. It's being released in January.

I also firmly want multiple paths/ways to overcome obstacles or navigate the world, like Deus Ex tried to do.

For example, to get through a locked door one might pick a lock, find the key, bash it open by brute strength, blast it open with magic or items, or find your way around. Having multiple "ways" of overcoming an obstacle can (and should) cross categories such as intuition, game items, knowledge, reflexes, etc...